Connectedness as a Measure of Robustness

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    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
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