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# Connectedness as a Measure of Robustness

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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.

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### Connectedness as a Measure of Robustness

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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