Talk to the Kirksville Chapter of Sigma Xi that describes research on describing the vascular structure of networks of HUVEC cells. I also talk a little bit about Truman's mathematical biology program.

1. Introduction Graph Theory Cells Community
Connectedness As A Measure of Robustness
Dr. Jason Miller
Department of Mathematics
Truman State University
November 17, 2006
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

2. Introduction Graph Theory Cells Community
About the Talk
Introduction
1
Graph Theory
2
Vascular Networks
3
Research Communities
4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

3. Introduction Graph Theory Cells Community
What is Graph Theory?
Fundamental Objects
An abstract graph is made up of
nodes, and
edges that connect nodes.
Example
This is the complete graph on 5
nodes. Its nodes are most thoroughly
interconnected.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

4. Introduction Graph Theory Cells Community
What is Graph Theory?
Fundamental Objects
An abstract graph is made up of
nodes, and
edges that connect nodes.
Example
This is the complete graph on 5
nodes. Its nodes are most thoroughly
interconnected.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

5. Introduction Graph Theory Cells Community
What is Graph Theory?
Fundamental Objects
An abstract graph is made up of
nodes, and
edges that connect nodes.
Example
This is the complete graph on 5
nodes. Its nodes are most thoroughly
interconnected.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

6. Introduction Graph Theory Cells Community
What is Graph Theory?
Fundamental Objects
An abstract graph is made up of
nodes, and
edges that connect nodes.
Example
This is the complete graph on 5
nodes. Its nodes are most thoroughly
interconnected.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

7. Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,
sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internet
and edge represent neworking that connects the computers.
Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodes
represent intersections. Analysis of such a graph can illuminate
how vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

8. Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,
sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internet
and edge represent neworking that connects the computers.
Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodes
represent intersections. Analysis of such a graph can illuminate
how vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

9. Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,
sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internet
and edge represent neworking that connects the computers.
Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodes
represent intersections. Analysis of such a graph can illuminate
how vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

10. Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,
sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internet
and edge represent neworking that connects the computers.
Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodes
represent intersections. Analysis of such a graph can illuminate
how vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

11. Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,
sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internet
and edge represent neworking that connects the computers.
Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodes
represent intersections. Analysis of such a graph can illuminate
how vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

12. Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
My Interest
Graph connectedness is a measure of
1 robustness.
Example (Complete Graph, 5 Nodes)
2 5 Complete graphs are robust against
losing nodes.
Lose node #5, and the remaining
nodes and edges still form a single
network.
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

13. Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
My Interest
Graph connectedness is a measure of
1 robustness.
Example (Complete Graph, 5 Nodes)
2 5 Complete graphs are robust against
losing nodes.
Lose node #5, and the remaining
nodes and edges still form a single
network.
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

14. Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
My Interest
Graph connectedness is a measure of
1 robustness.
Example (Complete Graph, 5 Nodes)
2 5 Complete graphs are robust against
losing nodes.
Lose node #5, and the remaining
nodes and edges still form a single
network.
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

15. Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
My Interest
Graph connectedness is a measure of
1 robustness.
Example (Complete Graph, 5 Nodes)
2 5 Complete graphs are robust against
losing nodes.
Lose node #5, and the remaining
nodes and edges still form a single
network.
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

16. Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
My Interest
Graph connectedness is a measure of
1 robustness.
Example
2 5 This graph is not robust against
losing nodes.
Lose node #5, and the remaining
nodes and edges form two separate
networks.
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

17. Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
My Interest
Graph connectedness is a measure of
1 robustness.
Example
2 5 This graph is not robust against
losing nodes.
Lose node #5, and the remaining
nodes and edges form two separate
networks.
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

18. Introduction Graph Theory Cells Community
Theorems on Connectedness
A network structure can be encoded into a matrix using node
adjacency.
Definition (Adjacency Matrix)
The ijth entry of the n × n adjacency matrix A of a graph G is

 1 if i = j and the i th and jth nodes are
Aij = connected with an edge
0 otherwise

J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

19. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
 
0 1 1 1 1
1 0 1 1 1
2 5
 
 
A=  1 1 0 1 1 
 
1 1 1 0 1
 
1 1 1 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

20. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
 
0 1 1 1 1
1 0 1 1 1
2 5
 
 
A=  1 1 0 1 1 
 
1 1 1 0 1
 
1 1 1 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

21. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
 
0 1 1 1 1
1 0 1 1 1
2 5
 
 
A=  1 1 0 1 1 
 
1 1 1 0 1
 
1 1 1 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

22. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
 
0 1 1 1 1
1 0 1 1 1
2 5
 
 
A=  1 1 0 1 1 
 
1 1 1 0 1
 
1 1 1 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

23. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
 
0 1 1 1 1
1 0 1 1 1
2 5
 
 
A=  1 1 0 1 1 
 
1 1 1 0 1
 
1 1 1 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

24. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix)
1
 
0 1 0 0 1
1 0 0 0 1
2 5
 
 
A=  0 0 0 1 0 
 
0 0 1 0 1
 
1 1 0 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

25. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix)
1
 
0 1 0 0 1
1 0 0 0 1
2 5
 
 
A=  0 0 0 1 0 
 
0 0 1 0 1
 
1 1 0 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

26. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix)
1
 
0 1 0 0 1
1 0 0 0 1
2 5
 
 
A=  0 0 0 1 0 
 
0 0 1 0 1
 
1 1 0 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

27. Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix)
1
 
0 1 0 0 1
1 0 0 0 1
2 5
 
 
A=  0 0 0 1 0 
 
0 0 1 0 1
 
1 1 0 1 0
(Note: i → column, j → row)
3 4
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

28. Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of the
graph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly into
another matrix call a Laplacian matrix whose eigenvalues and
eigenvectors give structural information about the graph. We hope
to exploit this information to describe the robustness of vascular
networks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

29. Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of the
graph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly into
another matrix call a Laplacian matrix whose eigenvalues and
eigenvectors give structural information about the graph. We hope
to exploit this information to describe the robustness of vascular
networks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

30. Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of the
graph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly into
another matrix call a Laplacian matrix whose eigenvalues and
eigenvectors give structural information about the graph. We hope
to exploit this information to describe the robustness of vascular
networks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

31. Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of the
graph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly into
another matrix call a Laplacian matrix whose eigenvalues and
eigenvectors give structural information about the graph. We hope
to exploit this information to describe the robustness of vascular
networks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

32. Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of the
graph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly into
another matrix call a Laplacian matrix whose eigenvalues and
eigenvectors give structural information about the graph. We hope
to exploit this information to describe the robustness of vascular
networks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

33. Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.
Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients through
diffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vessels
near to or inside the tumor. (Some attract host vessel, others
create their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? How
can they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

34. Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.
Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients through
diffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vessels
near to or inside the tumor. (Some attract host vessel, others
create their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? How
can they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

35. Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.
Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients through
diffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vessels
near to or inside the tumor. (Some attract host vessel, others
create their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? How
can they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

36. Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.
Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients through
diffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vessels
near to or inside the tumor. (Some attract host vessel, others
create their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? How
can they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

37. Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.
Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients through
diffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vessels
near to or inside the tumor. (Some attract host vessel, others
create their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? How
can they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

38. Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.
Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients through
diffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vessels
near to or inside the tumor. (Some attract host vessel, others
create their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? How
can they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

39. Introduction Graph Theory Cells Community
Vascular Networks
Background: Angiogenesis
Vessel formation can be good, too.
Example
Wounds heal.
Example
Blood flow reroutes when vessels are blocked (e.g., stroke).
Big Question
What are some of the mechanisms at work that allow this? How
can they be promoted?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

40. Introduction Graph Theory Cells Community
Vascular Networks
Background: Angiogenesis
Vessel formation can be good, too.
Example
Wounds heal.
Example
Blood flow reroutes when vessels are blocked (e.g., stroke).
Big Question
What are some of the mechanisms at work that allow this? How
can they be promoted?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

41. Introduction Graph Theory Cells Community
Vascular Networks
Background: Angiogenesis
Vessel formation can be good, too.
Example
Wounds heal.
Example
Blood flow reroutes when vessels are blocked (e.g., stroke).
Big Question
What are some of the mechanisms at work that allow this? How
can they be promoted?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

42. Introduction Graph Theory Cells Community
Vascular Networks
Background: Angiogenesis
Vessel formation can be good, too.
Example
Wounds heal.
Example
Blood flow reroutes when vessels are blocked (e.g., stroke).
Big Question
What are some of the mechanisms at work that allow this? How
can they be promoted?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

43. Introduction Graph Theory Cells Community
Vascular Networks
Research Project
Question
How can we effectively measure the effects of promoting or
inhibiting vasculogenic or angiogenic processes?
This is a question posed to a group of faculty and
undergraduates in 2004 by Robert Baer.
Example (Model system)
Human umbilical vein endothelial cells (HUVEC) self organize into
networks of vessels.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

44. Introduction Graph Theory Cells Community
Vascular Networks
Research Project
Question
How can we effectively measure the effects of promoting or
inhibiting vasculogenic or angiogenic processes?
This is a question posed to a group of faculty and
undergraduates in 2004 by Robert Baer.
Example (Model system)
Human umbilical vein endothelial cells (HUVEC) self organize into
networks of vessels.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

45. Introduction Graph Theory Cells Community
Vascular Networks
Research Project
Question
How can we effectively measure the effects of promoting or
inhibiting vasculogenic or angiogenic processes?
This is a question posed to a group of faculty and
undergraduates in 2004 by Robert Baer.
Example (Model system)
Human umbilical vein endothelial cells (HUVEC) self organize into
networks of vessels.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

46. Introduction Graph Theory Cells Community
Vascular Networks
Mathematical Biology Initiative, summer 2004
An NSF training grant in mathematical biology allowed this group
to take an image analytic approach to this question.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

47. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
number of junctions
network length
network area
number of meshes
size of meshes
Computer Aided Analysis
How can we get a computer to make these measurements
effectively with a minimum of human direction?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

48. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
number of junctions
network length
network area
number of meshes
size of meshes
Computer Aided Analysis
How can we get a computer to make these measurements
effectively with a minimum of human direction?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

49. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature
(view 1)
medial axis
meshes
segmented vasculature
(view 2)
medial information,
nodes
medial graph
newtwork
representation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

50. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature
(view 1)
medial axis
meshes
segmented vasculature
(view 2)
medial information,
nodes
medial graph
newtwork
representation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

51. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature
(view 1)
medial axis
meshes
segmented vasculature
(view 2)
medial information,
nodes
medial graph
newtwork
representation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

52. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature
(view 1)
medial axis
meshes
segmented vasculature
(view 2)
medial information,
nodes
medial graph
newtwork
representation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

53. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature
(view 1)
medial axis
meshes
segmented vasculature
(view 2)
medial information,
nodes
medial graph
newtwork
representation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

54. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature
(view 1)
medial axis
meshes
segmented vasculature
(view 2)
medial information,
nodes
medial graph
newtwork
representation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

55. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature
(view 1)
medial axis
meshes
segmented vasculature
(view 2)
medial information,
nodes
medial graph
newtwork
representation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

56. Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature
(view 1)
medial axis
meshes
segmented vasculature
(view 2)
medial information,
nodes
medial graph
newtwork
representation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

57. Introduction Graph Theory Cells Community
Research Groups
Mathematical Biology Initiative, summer 2004
At the same time in 2004, another research group was supported
by the same NSF training grant - statistical habitat suitability
model for Lesquerella filiformis (the MO Bladder-pod).
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

58. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved into
something bigger.
Biweekly Mathematical Biology Seminar, — a life science
fashion show
Connected more research active biology faculty with more
talented mathematics faculty
Supported the evolution of faculty scholarship in math and
biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

59. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved into
something bigger.
Biweekly Mathematical Biology Seminar, — a life science
fashion show
Connected more research active biology faculty with more
talented mathematics faculty
Supported the evolution of faculty scholarship in math and
biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

60. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved into
something bigger.
Biweekly Mathematical Biology Seminar, — a life science
fashion show
Connected more research active biology faculty with more
talented mathematics faculty
Supported the evolution of faculty scholarship in math and
biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

61. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved into
something bigger.
Biweekly Mathematical Biology Seminar, — a life science
fashion show
Connected more research active biology faculty with more
talented mathematics faculty
Supported the evolution of faculty scholarship in math and
biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

62. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved into
something bigger.
Biweekly Mathematical Biology Seminar, — a life science
fashion show
Connected more research active biology faculty with more
talented mathematics faculty
Supported the evolution of faculty scholarship in math and
biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

63. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

64. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

65. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

66. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

67. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

68. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

69. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

70. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

71. Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals with
intensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) at
national and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
Currently, over 9 biology faculty, 10 math & cs faculty, and 3 other
faculty are actively involved in this community.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

72. Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to use
undergraduate research as a way to
expand the STEM talent pool through high-quality
undergraduate research experiences
bring together research faculty in all STEM areas into a single
summer community
foster faculty scholarship
Community
Together, the Next STEP and MathBio programs have
dramatically increased the connections between faculty and
students of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

73. Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to use
undergraduate research as a way to
expand the STEM talent pool through high-quality
undergraduate research experiences
bring together research faculty in all STEM areas into a single
summer community
foster faculty scholarship
Community
Together, the Next STEP and MathBio programs have
dramatically increased the connections between faculty and
students of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

74. Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to use
undergraduate research as a way to
expand the STEM talent pool through high-quality
undergraduate research experiences
bring together research faculty in all STEM areas into a single
summer community
foster faculty scholarship
Community
Together, the Next STEP and MathBio programs have
dramatically increased the connections between faculty and
students of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

75. Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to use
undergraduate research as a way to
expand the STEM talent pool through high-quality
undergraduate research experiences
bring together research faculty in all STEM areas into a single
summer community
foster faculty scholarship
Community
Together, the Next STEP and MathBio programs have
dramatically increased the connections between faculty and
students of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

76. Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to use
undergraduate research as a way to
expand the STEM talent pool through high-quality
undergraduate research experiences
bring together research faculty in all STEM areas into a single
summer community
foster faculty scholarship
This is Truman’s “The Next STEP” program.
Community
Together, the Next STEP and MathBio programs have
dramatically increased the connections between faculty and
students of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

77. Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to use
undergraduate research as a way to
expand the STEM talent pool through high-quality
undergraduate research experiences
bring together research faculty in all STEM areas into a single
summer community
foster faculty scholarship
This is Truman’s “The Next STEP” program.
Community
Together, the Next STEP and MathBio programs have
dramatically increased the connections between faculty and
students of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

78. Introduction Graph Theory Cells Community
Community
Challenges
Sustainability
Conversion of student collaborations to peer reviewed work
Supporting continued faculty scholarship and research
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

79. Introduction Graph Theory Cells Community
Community
Challenges
Sustainability
Conversion of student collaborations to peer reviewed work
Supporting continued faculty scholarship and research
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

80. Introduction Graph Theory Cells Community
Community
Challenges
Sustainability
Conversion of student collaborations to peer reviewed work
Supporting continued faculty scholarship and research
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

81. Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and are
helping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, and
joyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm for
learning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

82. Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and are
helping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, and
joyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm for
learning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

83. Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and are
helping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, and
joyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm for
learning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

84. Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and are
helping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, and
joyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm for
learning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness

85. Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and are
helping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, and
joyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm for
learning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness