Dynamic Range and Dynamic Range Processors. Relative Sound Level Graph.
A dynamic graph model of kidney exchange
description
Transcript of A dynamic graph model of kidney exchange
![Page 1: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/1.jpg)
A dynamic graph model of kidney exchange
Yashodhan KanoriaMicrosoft Research New England & Columbia
Joint work with Ross Anderson, Itai Ashlagi and David Gamarnik
MIT
![Page 2: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/2.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Over 90,000 patients on the waiting list for cadaver kidneys in the U.S. today
In 2011:• 33,581 patients were added to the kidney waiting
list, and 28,625 patients were removed from the list.
• 11,043 transplants of cadaver kidneys performed.• 4,697 patients died while on the waiting list and
2,466 others removed from the list as “Too Sick to Transplant”.
• 5,771 transplants of kidneys from living donors in the US.
Kidney transplants
![Page 3: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/3.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Donor 1Blood type X
Recipient 1Blood type Y
Donor 2Blood type Y
Recipient 2Blood type X
2-way kidney exchange
Kidney exchange
![Page 4: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/4.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Donor 1 Recipient 1
Donor 2 Recipient 2Donor 3 Recipient 3
Pair 3
3-pair exchange (6 simultaneous surgeries)
Pair 1
Pair 2
![Page 5: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/5.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Compatibility graph
1 2 3
5
7
6
8
4 9
![Page 6: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/6.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• 4-way and larger exchanges have been successfully demonstrated
• However, significant challenges in conducting very large exchanges
Multi-way exchanges
![Page 7: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/7.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Question: Suppose only -way or smaller exchanges are possible. • Greedy policy: Complete an exchange as soon
as possible• Batch policy: Wait for many nodes to arrive
and then ‘pack’ exchanges optimally in compatibility graph
Which policy works better?
![Page 8: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/8.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Suppose, all donor-patient pairs have same probability of being compatible nodes form directed Erdos-Renyi graph.
Graph-structured queuing system:• At each time , a node arrives• Node forms edge with each node in the
system independently with probability • If cycle of size is formed, it may be eliminated
Objective: Minimize average waiting time =
Average(#nodes in system)Call this
![Page 9: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/9.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
If , then easy to achieve average waiting time
• But hospitals withhold easy to match pairs from exchanges (Ashlagi et al. 2011)
• Result: patient-donor pools presently consist of hard to match pairs
We consider
![Page 10: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/10.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• Two-cycle formed between any two nodes w.p. • Greedy exchange achieves • Not hard to show that for any policy • Hence, greedy achieves order optimal
Proposition: Greedy is optimal up to constants for
Only two-cycles:
![Page 11: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/11.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
What about
![Page 12: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/12.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• If batch size is then • We want to eliminate most of batch, so triangles
needed• Hence, need
Can show that batch size gives
How does greedy compare?
Batching for
![Page 13: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/13.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• Greedy removes 2 & 3 cycles as soon as available
• For a typical time , number of waiting nodes • Residual graph contains no 2 & 3 cycles, less
dense than ER• Optimistically contains 2 edges
Greedy for
![Page 14: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/14.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• Residual graph optimistically contains 2 edges • Probability that 2 or 3-cycle formed is in steady
state• Probability of 3-cycle formation ~
Need to make this • Probability of 2-cycle formation ~
Need to make this • So 3-cycle formation dominates, and ,
heuristicallySeems like greedy may not do to badly
Greedy for
![Page 15: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/15.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
1 2 4 8 16 32 62 1280
10
20
30
40
50
60
70
Size of batch
W
Simulation results: p = 0.081𝑝1.5 ≈44.2
![Page 16: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/16.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Simulation results: p = 0.05
1 2 4 8 16 32 62 1280
20
40
60
80
100
120
Size of batch
W
1𝑝1.5 ≈90
![Page 17: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/17.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Simulation results: p = 0.02
1 2 4 8 16 32 62 128200
210
220
230
240
250
260
270
280
Size of batch
W
1𝑝1.5 ≈350
![Page 18: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/18.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Optimal batch size is 1 (i.e., greedy beats batching)
Under greedy for small
What can we prove?
Summary of simulation results
![Page 19: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/19.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• Batching with maximal packing of cycles is monotone
• Shows that greedy is optimal up to a constant factor
Open problem: get rid of the constant factor slack, and consider all possible policies
Main result
Theorem: For we have• Greedy achieves • For any monotone policy
![Page 20: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/20.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• Suppose nodes in the system at • Want to show negative drift over next few time
steps• Worst case is emptyConsider next arrivals. For appropriate show:• Few new arrivals persist till • Few triangles formed internal to new arrivals• So most new arrivals form cycles containing
old nodes, leading to, whp,
Proof idea: greedy is good
![Page 21: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/21.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Consider graph of compatibility G between all nodes that ever arrive to the system.
A policy is monotone if:Fix all edges in G except for . Presence of only makes and disappear (weakly)
earlier.
Definition: monotone policies
![Page 22: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/22.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• Proof by contradiction. Assume .• w.p. at least . Assume this.• Under monotone policy, • Probability of immediate triangle formation for node is• Whp, no more than edges formed between and .
Assume this.• Probability forms triangle with next arrivals • With probability node lives longer than
Proof that no monotone policy can beat greedy
![Page 23: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/23.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
We analyzed a dynamic graph/graph structured queue:showed that greedy is nearly optimal. Suggests that greedy should work well in kidney exchanges.Caveats:• Greedy proved optimal only up to constant factors• Only consider monotone policies
Conclusion
Conjecture: For greedy gives ,and no policy can do better.
![Page 24: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/24.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• General result on ER-type graph structured queues with removal of given constant sized substructures?
• Kidneys: Multitype model with only some hard-to-match patients?Can we do better than greedy?
Future work
![Page 25: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/25.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Thank you!
![Page 26: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/26.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Pair 1
Pair 2
Pair 3
Pair 4
Pair 6
Pair 7
Pair 5
Altruistic donor
Altruistic donors: cycles plus chains
![Page 27: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/27.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
![Page 28: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/28.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• One altruistic donor at every stage(initially a volunteer, later a donor whose patient already got a kidney)
• A node arrives at each forms link with each existing node independently with probability
• Can eliminate any chain starting with altruistic donor. Last node in chain becomes new altruistic donor
Question: What is the optimal policy? Greedy or batch?
Model
![Page 29: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/29.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• d
Batch produces matching upper bound
![Page 30: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/30.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Ongoing work: what about greedy?
![Page 31: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/31.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Future work
![Page 32: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/32.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Pair 1
Pair 2
Pair 3
Pair 4
Pair 6
Pair 7
Pair 5
Altruistic donor
Altruistic donors: cycles plus chains
33
![Page 33: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/33.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Previous efficiency results
37
In a really large market efficiency is gained with short cycles:
Roth, Sonmez & Ünver, AER 2007 – if there are no tissue type incompatibilities, no need for exchanges of size >4
Ünver, ReStud 2009 - efficient dynamic kidney exchange assuming no tissue type incompatibilities - exchanges of size > 4 are not needed
Ashlagi & Roth 2010, in large random exchange pools, no need for exchanges of size>3
Toulis and Parkes 2011, similar results
![Page 34: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/34.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
n hospitals, each of a size c>0 D(n) - random compatibility graph:1. n pairs/nodes are randomized –compatible pairs are
disregarded2. Edges (crossmatches) are randomized
Random graphs will allow us to ask two related questions:What would efficient matches look like in an “ideal”
large world?
Random Compatibility Graphs
38
![Page 35: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/35.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Theorem (Erdos-Renyi) G(n,p) contains a perfect matching with probability approaching 1 as n grows for even n when p>log n/n.
- Random graph on n nodes with edge probability p
“Proof”: Say . Use Use greedy algorithm. Probability of failure in step k is
As long as Probability of failure at any step is
Matchings in random graphs
![Page 36: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/36.jpg)
B-A
B-AB A-AB
VA-B
A-O B-O
AB-O
O-B O-A
A-B
AB-B
AB-A
O-AB
O-OA-A B-
B AB-
AB
Efficiency in a large poolTheorem (Ashlagi and Roth, 2011): In almost every large random graph (directed edges are created with probability p) there is an efficient allocation with exchanges of size at most 3.
“Under-demanded” pairs
![Page 37: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/37.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Non-simultaneous extended altruistic donor chains (reduced risk from a broken link)
41
A. Conventional 2-way Matching
R1 R2
D1 D2
B. NEAD Chain Matching
R1 R2
D1 D2LND
A. Conventional 2-way Matching
R1 R2
D1 D2
R1 R2
D1 D2
B. NEAD Chain Matching
R1 R2
D1 D2LND
B. NEAD Chain Matching
R1 R2
D1 D2LND
Since non-directed donor chains don’t require simultaneity, they can be longer…
![Page 38: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/38.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
The First NEAD Chain (Rees, APD)
42
Recipient PRA
* This recipient required desensitization to Blood Group (AHG Titer of 1/8).# This recipient required desensitization to HLA DSA by T and B cell flow cytometry.
MI
O
AZ
July2007
O
O
62
1
Cauc
OH
July2007
A
O
0
2
Cauc
OH
Sept2007
A
A
23
3
Cauc
OH
Sept2007
B
A
0
4
Cauc
MD
Feb2008
A
B
100
5
Cauc
MD
Feb2008
A
A
64
7
Cauc
NC
Feb2008
AB
A
3
8
Cauc
OH
March2008
AB
A
46
10
AA Recipient Ethnicity
MD
Feb2008
A
A
78
6
Hisp
# *
MD
March2008
A
A
100
9
Cauc
HusbandWife
MotherDaughter
DaughterMother
SisterBrother
WifeHusband
FatherDaughter
HusbandWife
FriendFriend
BrotherBrother
DaughterMother
Relationship
![Page 39: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/39.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
![Page 40: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/40.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
In a really large market efficiency is gained with short cycles…
Are NEAD chains effective?
44
![Page 41: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/41.jpg)
B-A
B-AB A-AB
VA-B
A-O B-O
AB-O
O-B O-A
A-B
AB-B
AB-A
O-AB
O-OA-A B-
B AB-
AB
Efficiency in a large pool
altruistic donorAn altruistic donor can increase the
match size by at most 3
![Page 42: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/42.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• The large graph model with constant p (for each kind of patient-donor pair) predicts that only short chains are useful.
• But we now see long chains in practice.• They could be inefficient—i.e. competing with
short cycles for the same transplants.• But this isn’t the the case when we examine the
data.
A disconnect between model and data:
46
![Page 43: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/43.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
We have formulated and solved on real data
One donor added
Long cycles and altruistic donors help!
![Page 44: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/44.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Why? many very highly sensitized patients
48
Previous simulations: sample a patient and donor from the general population, discard if compatible (simple live transplant), keep if incompatible. This yields 13% High PRA.
The much higher observed percentage of high PRA patients means compatibility graphs will be sparse
![Page 45: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/45.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
PRA distribution in historical data
49
PRA – “probability” for a patient to pass a cross-match test (tissue type) with a random donor
![Page 46: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/46.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Short cycles leave many highly sensitized patients unmatched
![Page 47: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/47.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
A real graph
Graph induced by pairs with A patients and A donors 38 pairs, only 5 can be covered by some cycle
![Page 48: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/48.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Jellyfish structure of the compatibility graph: highly connected low sensitized pairs, sparse hi-
sensitized pairs
52
![Page 49: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/49.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Cycles and paths in random dense-sparse graphs
• n nodes. Each node is L w.p. q<1/2 and H w.p. 1-q
• incoming edges to L are drawn w.p.
• incoming edges to L are drawn w.p.
L
H
53
![Page 50: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/50.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Cycles and paths in random sparse (sub)graphs (v=0, only highly sensitized patients)
H
Theorem. (a) The number of cycles of length O(1) is O(1). (b) But when pH is a large constant there is cycle with length O(n)
“Proof” (a):
54To be logistically feasible, a long cycle must be a chain, i.e. contain a NDD
![Page 51: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/51.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Cycles and paths in random sparse graphs (v=0)
H
Theorem. (a) The number of cycles of length O(1) is O(1). (b) But when pH is a large constant there is path with length O(n)
Since cycles need to be short (as they need to be conducted simultaneously) but chains can be long (as they can be initiated by an altruistic donor,) the value of a non-directed donor is very large!
55
![Page 52: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/52.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Case v>0. Why increasing cycle bound helps?
L
H
Theorem. Let Ck be the largest number of transplants achievable with cycles · k. Let Dk be the largest number of transplants achievable with cycles · k plus one altruistic donor. Then for every constant k there exists ½>0
Furthermore, Ck and Dk cover almost all L nodes.
![Page 53: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/53.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Fact: in almost every random directed random graph D(n,c/n) every tree of constant size appears linearly many times and there are no constant size cycles
Lemma: Let p(n)=c/n. Almost every random bipartite graph G(qn,(1-q)n,p(n)) has a maximum matching of a linear size z(c,q)qn, 0<z(c,q)<1
Some more on random graphs
57
qn nodes
(1-À)n nodes
![Page 54: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/54.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Definition: u,v1,v2,…,vk is a good cycle if:• u is L and all other nodes are H• the only L node that has an edge to v1,v2,…,vk is u• the only H node that u has an edge to is v1
• No edges from v1,v2,…,vk to other H nodes• No edges from v2,…,vk to u
Claim: there are linearly many good cycles of length k+1
L
Hv1
v2v3
u
![Page 55: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/55.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graph
k=3
![Page 56: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/56.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graph
k=3
![Page 57: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/57.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graphStep 2: find maximum number of disjoint edges from L to beginnings of paths
k=3
![Page 58: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/58.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graphStep 2: find maximum number of disjoint edges from L to beginnings of paths
k=3
![Page 59: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/59.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graphStep 2: find maximum number of disjoint edges from L to beginnings of pathsStep 3: there is a linear number of edges that close good cycles from the last nodes of the established paths
k=3
![Page 60: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/60.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
Claim: there are linearly many good cycles of length k+1Step 1: there are linearly many isolated paths of length k in the H graphStep 2: find maximum number of disjoint edges from L to beginnings of pathsStep 3: there is a linear number of edges that close good cycles from the last nodes of the established paths
k=3
![Page 61: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/61.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
So far: linearly many good cycles of length k+1 Final step: Start from an allocation Qk and construct from it a Qk+1
allocation that adds a linear term (using the good cycles of length k+1)
k=3
![Page 62: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/62.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
So far: linearly many good cycles of length k+1 Final step: Start from an allocation Qk and construct from it a Qk+1
allocation that adds a linear term (using the good cycles of length k+1)
Add a good cycle if it is disjoint from Qk or delete the cycle that contains u in Qk and add it …
k=3
v2 v3v1
u
![Page 63: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/63.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
So far: linearly many good cycles of length k+1 Final step: Start from an allocation Qk and construct from it a Qk+1
allocation that adds a linear term (using the good cycles of length k+1)
Add a good cycle if it is disjoint from Qk or delete the cycle that contains u in Qk and add it …
k=3
v2 v3v1
u
![Page 64: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/64.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
L
H
So far: linearly many good cycles of length k+1 Final step: Start from an allocation Qk and construct from it a Qk+1
allocation that adds a linear term (using the good cycles of length k+1)
Add a good cycle if it is disjoint from Qk or delete the cycle that contains u in Qk and add it …
k=3
v2 v3v1
u
![Page 65: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/65.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Long chains benefit highly sensitized patients (without harming low-sensitized patients)
69
![Page 66: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/66.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
NYTimes February 18, 2012. 60 lives, 30 kidneys
![Page 67: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/67.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
What is the tradeoff between waiting and number of matches?
Dynamic matching in dense graphs (Unver, ReStud,2010).
What about dynamics?
![Page 68: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/68.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Matching over time
72
Simulation results using 2 year data from NKR*
In order to gain in current pools, we need to wait probably “too” long
*On average 1 pair every 2 days arrived over the two years
1 5 10 20 32 64 100 260 520 1041300
350
400
450
500
550
2-ways3-ways2-ways & chain3-ways & chain
Waiting period between match runs
Matches
![Page 69: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/69.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Matching over time
73
Simulation results using 2 year data from NKR*
In order to gain in current pools, we need to wait probably “too” long
*On average 1 pair every 2 days arrived over the two years
Matches – high PRA
1 5 10 20 32 64 100 260 520 104190
110
130
150
170
190
210
230
2-ways3-ways2-ways & chain3-ways & chain
![Page 70: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/70.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Matching over time
74
1D 1W 2W 1M 3M 6M 1Y250255260265270275280285290295
Matches
Simulation results using 2 year data from NKR*
1D 1W 2W 1M 3M 6M 1Y100120140160180200220240
Waiting Time
In order to gain in current pools, we need to wait probably “too” long
*On average 1 pair every 2 days arrived over the two years
![Page 71: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/71.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Matching over time
75
Simulation results using 2 year data from NKR
1D 1W 2W 1M 3M 6M 1Y4045505560657075
Matches – high PRA
1D 1W 2W 1M 3M 6M 1Y210
230
250
270
290
Waiting – high PRA
*On average 1 pair every 2 days arrived over the two years
![Page 72: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/72.jpg)
Match the pair right away?
A H-node forms an edge with each node u of U with probability ξ/n. A L-node forms an edge with each node u of U with probability π
76
Arriving pair
Lemma: the online algorithm matches almost all pairs when p is a constant and n is large enough (even with just 2-way cycles)
Online:match the arrived node to a neighbor; remove cycles when
formed.
Either a sparse finite horizon modelor an infinite horizon model and analyze steady state
![Page 73: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/73.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Dynamic matching in dense-sparse graphs
• n nodes. Each node is L w.p. q<1/2 and H w.p. 1-q
• incoming edges to L are drawn w.p.
• incoming edges to L are drawn w.p.
L
H
77
![Page 74: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/74.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Dynamic matching in dense-sparse graphs
• n nodes. Each node is L w.p. q<1/2 and H w.p. 1-q
• incoming edges to L are drawn w.p.
• incoming edges to L are drawn w.p.
L
H
78
At each time step 1,2,…, n, one node arrives.
![Page 75: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/75.jpg)
Heterogeneous Dynamic Model (PRA). PRA determines the likelihood that a patient
cannot receive a kidney from a blood-type compatible donor.
PRA < 79: Low sensitivity patients (L-patients). 80 < PRA < 100: High sensitivity patients (H-
patients). Most blood-type compatible pairs that join the pool have H-patients. Distribution of High PRA patients in the pool is different from the
population PRA.
79
pc/n
𝑝2
![Page 76: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/76.jpg)
Chunk Matching in a heterogeneous graph
80
At time steps Δ, 2Δ, …, n:
Find maximum matching in H-L; remove the matched nodes.
Find maximum matching in L-L; remove the matched nodes.
![Page 77: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/77.jpg)
Chunk Matching in a heterogeneous graph
81
Theorem (Ashlagi, Jalliet and Manshadi): When matching only 2-way or 2+3-way cycles:
1. If Δ = o(n), M(Δ) = M(1) + o(n)
2. Δ = αn, then M(Δ) = M(1) + f(q)n
for strictly increasing f()>0.
Chunk matching finds a maximum matching at time steps Δ, 2Δ, …, n.M(Δ) - expected number of matched pairs at time n.
![Page 78: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/78.jpg)
Denser Poolsξ:
82
Theorem: 1. If Δ < 1/,
M(Δ) = M(1) + o(n)2. If Δ = α/
M(Δ) = M(1) + f(q)nfor strictly increasing f()>0.
Need to wait less time to gain… If the graph is dense (large) – no need to wait at all…
![Page 79: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/79.jpg)
Proof Ideas Special structure: Sparse H-L and dense L-L.
(PRA). PRA determines the likelihood that a patient cannot receive a kidney from a blood-
type compatible donor. PRA < 79: Low sensitivity patients (L-patients). 80 < PRA < 100: High sensitivity patients (H-patients).
Most blood-type compatible pairs that join the pool have H-patients. Distribution of High PRA patients in the pool is different from the population PRA.
Compare the number of H-L matchings.
83
pξ/n
𝑝2
![Page 80: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/80.jpg)
Proof IdeasIn H-L graph, Δ = o(n):
No edge in the residual graph.
Tissue-type compatibility: Percentage Reactive Antibodies (PRA). PRA determines the likelihood that a patient cannot receive a kidney from a blood-type
compatible donor. PRA < 79: Low sensitivity patients (L-patients). 80 < PRA < 100: High sensitivity patients (H-patients).
Most blood-type compatible pairs that join the pool have H-patients. Distribution of High PRA patients in the pool is different from the population PRA.
Decision of online and chunk matching are the same on depth-one trees. M(Δ) = M(1) + o(n).
84
arrived chunk
residual graph
![Page 81: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/81.jpg)
In H-L graph, Δ = αn: Find f(α)n augmenting paths to the matching obtained by online. Given M the matching of the online scheme:
Chunk matching would choose (l1,h1) and (l2,h2). M(Δ) = M(1) + f(α)n,
85
Proof Ideas
h1
l2 l1
h2
![Page 82: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/82.jpg)
Chunk Matching in a heterogeneous graph
86
Theorem (Ashlagi, Jalliet and Manshadi):
MC(1) = M(1) + f(q)n
Chunk matching finds a maximum matching at time steps Δ, 2Δ, …, n.M(Δ) - expected number of matched pairs at time n when matching only 2-ways MC(Δ) - expected number of matched pairs at time n when matching 2-ways and allowing one unbounded chain.
![Page 83: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/83.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Merging NKR and APD
5 10 20 50 100 250480500520540560580600
Pairs matched
5 10 20 50 100 250160
180
200
220
240
PRA >= 80 matched
5 10 20 50 100 2505060708090
100110120
PRA >= 97 matched
5 10 20 50 100 250202530354045505560
PRA >= 99 matched
![Page 84: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/84.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
• Current pools contain many highly sensitized patients and (long) chains are very effective.
(Partially since hospitals don’t share all their easy to match pairs.)• In those highly sensitized pools, number of
matches increase significantly only when waiting “for a long” time between match runs -> use more chains!
• Many more matches from pooling, especially highly sensitized patients.
Conclusions
89
![Page 85: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/85.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
Merging exchange programs
NKR Korea APD Korea SA Korea APD NKR APD SApairs 222 81 196 81 173 81 196 222 196 173
matched 30 16 49 16 59 16 49 30 49 59
Average PRA 61.9 7.1 57.8 7.1 59.1 7.1 57.8 61.9 57.8 59.1
PRA OD 75 43 85.6 43 75.2 43 85.6 75 85.6 75.2Pairs 192 65 147 65 114 65 147 192 147 114
matched 6,6 4,5 4,8 3,7 0,5 0,1 0,13 0,15 2,13 4,22
PRA>80 5,5 1,1 3,6 0,0 0,5 0,0 0,11 0,9 2,11 2,18OD 4,4 1,1 1,4 0,0 0,5 0,0 0,9 0,8 2,9 2,15O
donors 2,2 0,0 1,4 0,0 0,3 0,0 0,8 0,6 2,9 1,9
PRA OD 95,95 97,97 100,96.8 -,- -,98.8 -,- -,97.7 -,96.8 100,97.9 97.5,97.3
![Page 86: A dynamic graph model of kidney exchange](https://reader035.fdocuments.in/reader035/viewer/2022062302/56816368550346895dd44083/html5/thumbnails/86.jpg)
Yash Kanoria (MSR-NE) A dynamic graph model of kidney exchange
2000
2001
2002 2003 2004 2005 2006 2007 2008 2009
2010
#Kidney exchange transplants in US*
2 4 6 19 34 27 74 121 240 304 422 (+203 +139)*
Deceased donor waiting list (active + inactive) in thousands
54
56
59 61 65 68 73 78 83 88 89.9
Kidney exchange is progressing, but progress is still slow
*http://optn.transplant.hrsa.gov/latestData/rptData.asp Living Donor Transplants By Donor Relation•UNOS 2010: Paired exchange + anonymous (ndd?) + list exchange
In 2011: 11,043 transplants from deceased donors 5,769 transplants from living donors
s