had the chance to weigh it on this. One is your current work with Capstone, which I think is important in terms of Capstone clients and how they're affected by climate change and some of the things that we have in front of us, both in terms of the transition, in terms of the opportunity, in terms of some of the vulnerability of some of the folks who you serve. Also, you have kind of past professional experience that I think is really relevant to what we're doing here, in terms of your work as the chief recovery officer with regard to Tropical Storm. I read in terms of your work as the head of VTrans and the work that you did there, which is ultimately our transportation system is very closely tied into what's happening in the kind of climate emergency. So it was a variety of reasons. So I wanted to make sure that we had space and time for you to offer your thoughts on this bill, but also bring some of the things that are going on with Capstone right now, which we're eager to hear as well. Thank you for joining us. And as you know, we record our hearings for the records if you would mind introducing yourself. Absolutely. I really want to thank you for inviting me and kind of envision this maybe more as a discussion than sort of just me testifying. But for the record, I am Sue Minter, the executive director of Capstone Community Action, which is an anti-poverty organization in central Vermont, which really includes the counties of Orange, a few towns in Windsor, Washington County, and LaMoyle County. And Capstone is one of a network of community action agencies across the state, one of five.  I am here testifying really on behalf of myself and, as you say, some of my past experience. Good morning, Heidi. And I also, so I'm not testifying on behalf of VCAP, the network, but definitely representing Capstone. Thank you. And I wanted to just start out because driving here, I was listening to the radio as I tend to. And I learned that yesterday, which happened to have been my birthday, the doomsday clock was moved. And now, the doomsday clock, you may recall, was really about the atomic energy. And it is put forward by the Atomic Energy, not commission, scientists. The bulletin of the atomic scientists. Anyway, we are now 100 seconds to midnight by in the view of these scientists. And that incorporates both the threat of nuclear war and of climate change. So what I heard on the radio this morning was it's time to wake up, which I thought was good, because I didn't necessarily get a great night's sleep. So I thought I'd wake up for you. But just a reminder that this is a global emergency, that I'm really pleased that you are taking seriously and thinking, I guess, across sectors long term and sort of short term. And I do want to, and as I point out in my testimony, I appreciate your bringing some of my past experience here. Secretary V-Trans also Irene. And I think when we are talking about resilience and planning, I think the experience we had in Irene is the closest we've come to really what severe intense extreme storm looks like. And I did experience or learn in that experience that it is lowest income folks who are really the most vulnerable. They're often located in the highest risk areas, in structures such as mobile homes that are the least resilient. And in addition, they have the hardest time recovering. So I appreciate many pieces of your bill. I'm really here to offer sort of friendly amendments and to support your efforts. I also had the experience when I was. So my experience is I was at the Transportation Agency as the deputy secretary when Irene hit. And during that first four month period, which we consider the emergency phase, it was really logistics and response. After that phase, we moved into what they call the recovery phase, both the short and long term recovery. And that's when I was appointed to become in charge of. I would say that we were incredibly poorly equipped and unprepared for Irene. And one of the reasons I'm so pleased that you are thinking in the way that this bill does by bringing together a council that is cross sector and longer term in its horizon is because I worry a lot. We learn so many lessons in Irene and from Irene. We try to record them. We did a little bit of structural change within our government to at least have a position and a little post within the Department of Emergency Management who's thinking about mitigation and resilience. And we are certainly more prepared simply because of what we learned during that disaster. But I hope that we can use this opportunity to incorporate some of those lessons learned. I did also have the really extraordinary opportunity when I was redeployed back to VTRANS. The State of Vermont was asked to join an initiative of the Obama administration, the task force on climate preparedness and resilience. So this was, and I was lucky enough to represent Vermont, really, our governor in that position, which meant I was joined by 26 other governors, mayors, county commissioners, tribal land leaders from across the country, all these different jurisdictions, and trying to really combine and coalesce our lessons learned into a report. It was an extraordinary year and a half experience. I was then co-chairing a subcommittee that really talked about sort of recommendations for FEMA. It worries me that all of these good lessons, I don't know where they sit anymore, it was impressive that the entire cabinet of the former president actually attended our convenions and that every single cabinet member had climate preparedness in one of the top things they're doing. So I worry that we're becoming increasingly unprepared. So I'm very pleased by key elements of your plan. I did learn a little bit at a forum a couple of nights ago that other states in our region are adopting similar approaches. I guess I start by just, in my testimony, saying I very much appreciate that you do include marginalized and disadvantaged Vermonters or their representatives in your thinking for exactly the reason that I shared and I hadn't realized that the IPPC report actually referenced the disproportionate impact on these communities. And especially because the folks who lack insurance and lack resources will be the ones who the state worries the most about and spends the most on if they are so inclined, which I think our state really was following Irene, the more that we can be prepared, the lower those costs are. So I think there's an economic argument, frankly, to becoming more prepared in a broad sense. I do really support incorporating some clear enforceable reduction requirements. I was a member of this body along with many others in the room, or at least some. Representative Sherman and I shared a seat on the Transportation Committee. And when we, I think, first adopted this legislation, I think we set forth goals, and I think it's hard to achieve goals when they aren't really staring you from the face. And I don't think we as a state have figured out good, constructive ways to measure where we are relative to where we want to be in a consistent way and thereby think about strategies to reduce. So I can't possibly think to testify about what the right targets are, but I've certainly seen and I did as a graduate student study environmental policy. And at that time, watched very serious positive impacts under the Clean Air Act by mandating emission reductions and with targets. And in certain states, and I certainly think of California, who have already started adopting greenhouse gas emission requirement reductions as part of their air quality infrastructure have really seen differences. So I support that. It's not going to be easy, but I think it's the right approach. One recommendation that I'd like to make about the council and is really in the structure and who you have as the chair. And again, I think this is my experience after Irene. So I would recommend that you actually have the Secretary of the Agency of Administration chairing that council. When I was the Irene Recovery Officer and I followed Neil Lunderville in that position, he was there in the most intense first four months. And he had been the Secretary of Administration. And he, in that experience, understood in a way that few others could, the levers of powered influence and how to conduct significant response. I learned then by being within the Agency of Administration just how much when you are a secretary of one agency to you are on a parallel, you are not above the others. And unless you have someone who has the authority to actually oversee those other agencies, I don't think it's effective. I learned that I was in the Irene Recovery Office, and I think Neil Lunderville specifically set it up this way because of his understanding, having been in that position, that the Secretary of the Agency of Administration oversees all of state government on behalf of the governor. It's separate from the governor's chief of staff and staff. It's really responsible for government. And I created, we created a multi-sector inter-agency essentially cabinet around Irene. And I think because I had the authority of the Secretary of Administration, I was able to be more effective. And that's why I suggest you consider that structure. The couple of people who I think you may want to add to your membership, the regional planning folks, VACTA, I think, are very important. I didn't see them, at least on the draft that I had. And they're not on there. Yeah. You know, after Irene and certainly, OK, I have to say, I have a degree in planning. I believe in planning. But I think what this council is about is trying to plan for our state. And we certainly relied tremendously on the regional planning commissions to undertake a lot on behalf of their member states. So that makes some sense to me. The other thing, I don't know whether it makes sense, but at least you should hear from the Department of Public Service. One of the recommendations we made after Irene that was not adopted was that the emergency management leader actually become higher up within the cabinet and a member of cabinet. Many other states do that. And those are states who are much more used to having emergencies to respond to. Can you just walk through it again? So right now, you have the agency of administration and then you have the Department of Public Safety. And within that, you have a division of emergency management. So that person and group is really the ones who understand and they have relationships with FEMA and they are schooled in that field of work, but also have a lot more practice in emergency response and drills. And they're just so far from the cabinet. And that is not the case in many other states I've served and elevating it. I had recommended that, but it wasn't adopted. But we did create a position, actually, which is still there. We had never had someone at that time really full-time focused on mitigation and resilience planning. This is pre-2010. Pre-2011, yeah. We have a very, no, we're frugal. Let's put it that way. And we haven't had to respond to emergencies of that scale had we ever before. And we were, I would say, under-resourced in that area and not nearly as educated as many of our neighbor states in everything that is expected of us by FEMA, by our federal agencies. By contrast, I was running the Agency of Transportation. Our federal agency was the Federal Highway Administration. We are very, I would say, well-resourced in the Agency of Transportation relative to many other state agencies and departments in Vermont. And I think it has a lot to do with the fact that we have federal dollars and that we understand and that the transportation world understands what it takes to do the incredibly important work. But because of our partnership with Federal Highway, they were right with us every step of the way. We were interagency. So the damage that was done, and there were 600 miles of roads completely wiped out overnight. The damage that was done to the state highway had federal resources and an entire infrastructure to support how did we manage that and an entirely different way to get emergency funding. But significantly, so and then all of the infrastructure that the towns, the culverts and roads, let alone the buildings and homes and business loss, but just the infrastructure, for example, totally different federal agencies, totally different practices and standards. And we were able to implement very different standards in the way we responded, the size of the bridges and culverts that we built after Irene, the Federal Highway administration supported. They went out, we went out on tours together, understanding why we were doing this. It was a much larger investment and they supported and funded it. I mean, we had our battles, but they were agreed to. The challenge after Irene was with FEMA and every one of our towns, especially in some parts of the state that were desperately tiny towns impacted by eight times the cost of their annual, full annual budget. But the FEMA organization said, no, we're not paying for larger culverts. And we got into a situation where we actually, the government, we adopted larger codes and standards that we required the towns to use. And FEMA said, we're not paying for that. So literally for two years, we battled, legal battles and FEMA does not respond to a court. You can only appeal inside FEMA. So we had appeals to the local level and then we went down to Boston and then we went to Washington. And it went beyond that and thankfully because we have a senator, two senators with very strong relationships at that time right up to the president. But we had advocates going to the head of FEMA and we had calls to the president of the United States in order to get the dollars for our towns, for our state hospital, for our state complex, for major things. It was a political battle, sadly. But at the very basic core, we didn't have the ability to offer our towns and even FEMA the kind of information and training and understanding of what's expected of you. So that's what I see coming out of this, hopefully. A lot of lessons were learned. We do have people now who are trained in that. For our state government, I don't think we are probably resourced enough at our regional planning commissions and elsewhere to continue that work. Just as one example, now there are some plans that you require here. Every town is supposed to do a mitigation plan. When FEMA had almost, nobody had done it. And so we very quickly, we couldn't start getting the dollars flowing until they had adopted and approved these plans and it was a nightmare. So it's the kind of work that must happen for us to get expeditious dollars. Now, I worry if we were to suffer from a disaster at this time, when we don't necessarily have an administration who will kind of upon our state the way we did at that time. And I think the budgets are so much tighter than even they were then for disasters. I worry that the way in which FEMA was willing to wait and help may not be acceptable. So being prepared means a lot of things. Right down to every town having an approved disaster mitigation plan, which I think you'll get to. So can I ask a question to the person that got into emergency management? Yes, he's still there. And their role is? You'll have to ask, because I just remember, I think that the position's title is disaster mitigation and resilience or something. I'm sorry that I don't actually know, we called it something and I think it's still bad and they're doing a lot of work. When I spoke to emergency management this summer about this bill and where they're at and their capacity and one of the things that they had noted was they don't have as much capacity as they need to help towns get those hazard mitigation plans written because towns are a lot of volunteer towns, as you know. So do you know if it was that type of disability? Yes, it is. Disaster mitigation planning and preparedness. So there's all kinds of trainings that what does FEMA expect? I'm sorry that I don't have the details really in my head, but it is exactly that kind of work that is very basic that we were completely underprepared for before I green and we tried to make adjustments to help us be more prepared but at this moment I don't know where we sit. What was the difference in reimbursement from FEMA? Well it went literally town by town. I mean there's a lot of things. One of the first battles that we had to wage was getting to what they call the 90%. So instead of having the feds pay 75%, we got to that. So that was battle royale number one. But every single major investment was a battle and a negotiation and very crazy making. But literally every single, I mean I'll take the town of Bennington really extraordinary that they had, because the roaring brook had already flooded so many times, they had gotten federal funding, the FEMA funding, hazard mitigation funding to recover in a way that was resilient. So we've learned that you can't just build trenches. We have to think about the velocity of the moving water and so we know that rivers meander and so we try to replicate that in the way we now respond to flooding. And that's because of the really leading edge river fluvial geomorphologists that we have in our agency of natural resources with whom the transportation agency has also learned and we are really working together on how to respond in a more resilient fashion. And I would welcome you to invite Michelle Boomhauer and her team to share with you the resilience planning that the agency has done thanks to support from the federal highway administration. And I think you should learn about that because it's relevant. And I mentioned it in one of my recommendations, but the town of Bennington had a $2 million investment in how they properly responded in resilient fashion when the roaring brook was decimated and we had photographs and we had the geologists and we had, and FEMA said no, I'm not paying for that. So a $2 million for the town of Bennington is just an extraordinary impact. So we had to get the Senate, so Senator Leahy got some of the staffers to come and take a tour. I went to Washington and talked directly to the FEMA staff.  And actually a member of Michelle Boomhauer's team was at that whose now your environmental planner and policy was the attorney. You might be interested to get his testimony because it is telling what we had to do. We deployed attorneys from the Attorney General from the Agency of Transportation. We spent enormous resources to battle with FEMA on why they should pay for us to do exactly what the plans that the mitigation part of FEMA had said we should do. We won. Bennington got the money, the town, town by town, we battled on the culvert sizing. It's an extraordinary story and it is the mindset of FEMA not to be doing that. And I think it has to do with the fact that they don't have the money. Which I expect is exacerbated today. So however, the more prepared you are with your mitigation plans and the more you can, the other issue was mapping. We were relying on 20 year old maps from FEMA that were inundation maps and they weren't fluly old erosion maps. And what we now know is our threat is the river overflowing and the unbelievable damage that it will do. It is not, you know, slow-moving floods that then recede. And our Agency of Natural Resources had gotten funding from FEMA to do that kind of mapping and we had high hazards. So in the town of Jamaica, we had homes that were ripped away by that river. I think it was five homes and we could not get them funded for the buyout even though they met because they were on the high hazard map that our Agency of Natural Resources had already sent but they weren't on the FEMA inundation map. And it is just the lack of preparedness and planning and clarity that allows FEMA to say no until we battle back. So I went down a path I didn't expect to even talk to you about but it's relevant. And it's almost like PTSD to have these memories of the battle and it was very, very, very difficult. And it isn't that the people who work for FEMA are not very caring, hardworking people. I think they have sometimes got their hands tied behind their back and it's because we don't always have the funding available. And the battle for the funding unfortunately became a political battle and we were successful and it was a lot of work. So those were some of the things that I didn't expect to talk to you about. And I will just say, one of the other things I got to do was help just send a team, lead a team really of folks from transportation to Colorado after their flooding because their flooding was very similar to ours. And the whole transportation world and I was the head of something called NASTO, the Northeast Association of State Transportation Officials and we started a whole discussion around resilient infrastructure. So the world of transportation has learned a lot and is really moving forward on thinking differently about how we build and where we build. In Colorado, we recommended to them to have their recovery team also really tied at the governor's office at the highest level of state government and I think because of that, they like us I think were quite successful. I really can't say enough about the difference between being really in the top tier of government versus having a chair of a council that's really on the same level as every other of their cohorts. It's harder for everyone. I have a few more comments in there that you can read. I think I've sort of covered the main points and again, these were just friendly amendments with suggestions. Some is wording about adding social equity including public health and housing stock into a section that I think you didn't have. Making sure as you think about what's required that you are aware of the town requirement now and learning that it isn't well resourced doesn't surprise me but it does worry me. And that's what this council I think can help bring to light. So that probably would talk plenty. That's correct, I think that's a good question. So one of the pieces that we have in here is a creation of a vulnerability index to help us kind of understand which communities are most likely going to need help just with doing this planning or they're just going to be overwhelmed with that. We were surprised to find that the agency has a socioeconomic vulnerability index with state. I don't know if you were aware of that. You know, I didn't remember that. I think there's been a lot of post-Irene work that I haven't been tuned into but the Department of Health was involved with me and us on that Obama task force and there was a lot of good thinking and engagement by the health department at that time so I'm thrilled to hear that and that's very interesting and relevant. The mapping, again, the transportation agency has done a very good mapping of our infrastructure vulnerability relative to our river system, our fluvial hemorrhagy and has a prioritization system. I actually got to present that at a national climate resilience department it's a national transportation resilience conference I think it was a year ago. Maybe it was two years ago. I think it was two years ago. But anyway, we've done a lot of thinking with the support of the federal government to actually map our vulnerabilities and I think that all of the different mappings that may be going on may already be put together in some way but if they're not, that's a great thing and if they are, then that's the resource the council can begin to really utilize because while we're mapping and we even have a way of prioritizing I don't know if we've adopted those priorities and thought about the investments that need to be made and being strategic about getting going. That's what's not, I think. Reduce the dollar going. Yes, and in some ways it's about which bridge we've thought about redundancies when this bridge goes down, okay and so it's a very complicated formula of thinking about, okay, what do we need to do next and why, specifically to make ourselves stronger for the future. Can I keep going? You can, Mike wants to get in there too. Oh, okay, I'll wait. But if you want to continue on this. Well, I actually, around the, so one of the climate change activities that sometimes concerns me coming from rural areas and is thinking about cost-effective, right? So of course we need to do that, we need to be responsible but cost-effective when we're mitigating climate change really screams to me like let's get as few cars as possible in the cities, it screams to me of increasing pressure in the rural areas and so I'm wondering if you have given your experience of working in a lot of devastated rural areas. If you can, if you have any thoughts about how to balance that well. So thinking about cost-effective but also most vulnerable. I think the things that I think about the most with the rural areas is both the infrastructure and I know that we've done a lot of planning and thinking about that but it's where people are living relative to the flooding that's going to occur and the incredibly hard choice of suggesting that people move elsewhere and we did do that after Irene and this was another, that's what we had hoped to get federal money for in Jamaica but thankfully in Jamaica, the philanthropic community stepped in in a tremendous way to help those families but there are whole areas that we've mapped that we know really are higher risk and we should be talking about buyouts and some of the funding, we still have a pot of money and this is stuff you should learn more about and I'm not sure I get clear but we have mitigation funding sort of a percentage of every federally declared disaster a pot goes into a pot for thinking about how to utilize these funds. So probably there is good strategic thinking going on the Department of Environmental of Emergency Management is thinking about this stuff but I think a lot more about the folks at risk because of their homes than sort of where the cars I'm not sure what you mean about that but I do know that the folks who plan our infrastructure are thinking a lot about that and I'm confident that continues. But the housing and getting people to move or not build in high risk areas and that becomes the issue of insurance and that's another really difficult conversation. As we're dealing with, as we're looking to deal more aggressively with climate change and reduce our emissions but also protect our people dealing with the effects right now. So I wanna make sure that we don't set up a competing interest around reducing emissions and are protecting our folks that are dealing with climate change right now. Is that, maybe it's not, maybe it's not really clear. Sorry, I was going totally into the thinking of planning and resilience and not so much the impacts of mitigation and emission reductions. Well, I'm thinking a lot in my work at Capstone about different ways to invest in low income communities with how do we get people around differently? How do we, your body supported the effort to help reduce emissions by getting lower income folks helping them get subsidized to purchase used high efficiency vehicles. So getting off the road, what many, many low income people rely on, gas guzzling, high building vehicles but helping to subsidize them to get into the used high efficiency car market. So that's a very small example of something that last year you all passed in your T-bill that we're working with the Transportation Agency to implement and I'm hoping that that's a program that you'll continue to fund beyond this one year. It was a one time funds from the BW settlement so hopefully there will be a conversation and I'll be talking to others in the building about that and it's just one example and we are also working with the Agency for transportation on a very new program that the Federal Highway Administration is rolling out called Mobility for All. We have invested I think mightily for a rural state in our public transit system. I learned now with the folks that we serve that have access to no transportation and are literally trapped in poverty due to the lack of access. So we're looking to purchase or create an all electric fleet of vehicles to help get rides for people so they can go to work because our large employers can't find people and they're in our doors but they can't get there. People to get to their healthcare appointments which the hospitals are now very concerned of the number of people who aren't showing up. People get to their recovery. So we're trying to think locally about a system that uses electric transportation to meet the very desperate needs of folks in rural poverty. So thanks for your expertise. Your expertise is a goldmine of information on this. We're old and rusty about it. Gold never rests. Yeah, so, no, but I mean, it's amazing to hear the counsel what it took to get funding from FEMA. Anyway, where I was going to go though, as a director of FEMA, of Capstone, you do have programs that help people save money by reducing their greenhouse gas emissions and heating and transportation and such like that. Tell me about the backlog of people that are requesting assistance with weatherization and aren't able to get it. Do you have a lot of backlog there? Yes, we have definite waiting lists and we have waiting lists and we do not advertise. The only way that people know is honestly last year and I want to thank this committee. I know you really supported efforts to increase the amount of weatherization funding that we have and even just the debate, we got a spike in phone calls and that we can't really meet the demand. But part of what's challenging is in order to meet that demand, we have to hire new folks and we have to really build out our base of employees. And unless we know that that's gonna be a consistent and honestly, there's probably another expert in this room who could speak much more about this but it's a little bit, we wanna grow but we can't start hiring those folks unless we know there's a real growth path. I cannot testify exactly how many people are on a waiting list. I know that we're constantly having to triage and prioritize who it is we will serve first by a very complicated but important formula and I know we're not meeting the need. And one part of it is people up to 80% are in need of income are eligible but people below the 60% are prioritized so the people between 60 and 80 essentially never get served. Right, that's what I'm gonna be back to. So the only way to approach that is by increasing the decimal. Yes. Well we definitely, I do have, I hope just a four minute video maybe to close up and talk about a little bit of what we do. You have four minutes. I noticed that but I thank you really for inviting us into this room and even more so inviting representatives of the folks we serve onto this council and look forward to continuing the conversation. I'm happy to be a resource in any way. Yeah, I hope we get a chance to have you back. So this is just a video to give you a sense of what it is we do because what I've learned in my just over a year is that we have an extraordinary challenge of people in poverty who who are needing support whether it's food, heat, or housing that we do vital assistance for but that we actually can help people get out of poverty and that's the thing I feel is the most important work we do, we support people in crisis but we also have very innovative strategies to help people get economic and self-sufficiency. So this is a little portrayal of what we do. Thanks. My wife gets a check one week and I get one a couple weeks later and then I get one from the VA. It's just, you know, you've got to plan all your money for each painter. I didn't do a whole lot of good. I did a lot of drinking and I wasn't able to maintain jobs or relationships. I really didn't feel like I had much opportunity. A lot of people right now, their income is so limited and they really cannot make ends meet. I made the mistake of marrying to support for my family and got myself into a very bad relationship. I did 22 years in prison and I got out here. I didn't understand the seriousness of what I had for a fit. Not knowing that my life was going to be changed forever, that I wasn't going to ever raise my son or be a father of my child. I'm 71 years old and my wife and I are both retired and we're raising very children. So my parents just weren't able to raise their children and raise them properly. DCF stepped in and took them away from them and gave them to us. As a single mother, it was really hard. I lost apartments and I had to choose between feeding my kids or clothing them. I was homeless a lot of the time. We stayed in cars or couchserve. Without a second chance, what could we do besides going back to the old things that we used to do? We were struggling with, you know, we'd get a hand at three kids and financially just getting higher and higher and higher and higher. I wanted more. I wanted to open a business. I wanted to be more successful. I wanted a federal line for my son. The program is there. The opportunity is there. It's providing jobs for others. People want to help. So you take it, you do it. You run with it. It's an amazing program. It gave me the steps. It was somebody right there with me walking beside me to guide me through to say that I've now owned my business for 11 years. Capstone was like a key to a door I was trying to get through. They orchestrated the right moves and made me see that, OK, I can do this. I can be a cook for real. Whether it is the food shelf or the cooking class, I'll help at the times of this. This whole capstone is my foundation of support. When they did the furnace, it would save us about half of what our fuel was prior to here. I'm relieved that I'm going to be able to do more with the grandkids. I've come a long way and my children have seen it. And there's plenty of opportunity. And that's really what capstone has done. I mean, doesn't that be an American dream, really? To go be what you want and do what you want and be successful and be happy in your career. And capstone's given me that. I had hope to get out of prison, you know what I'm saying? And I made it. I hope to make it up here to Vermont. I did, you know, I hope to get a job. I got it, you know? Capstone gave people a way out of difficult situations. I say empowerment quite a bit because they do empower you to be better. And capstone has always been that stepping stone for me to get to that next level of confidence that I'm able to do so much more and help so many people because of capstone. I am president of Policy Council and part of the Head Start Program. With what I started with, which was not much, except for the desire to have my own business, then I was able to get that help, receive that help. It's fantastic. It's a team of things working together to accomplish one goal. And that goal is to live and to have better, and to work better. And to do better. You can't do that without a team. Thank you for listening. Nice to meet you. It's good work. Thank you. That guy's in the stuff. Yeah. Yeah, you've met him. Thanks for your work, everybody. I know it's hard. I know you're working hard and I appreciate it. Thank you for joining us early too. Yeah. It wasn't easy for you. Thank you for making time for us today. We're in the middle of considering the Global Warming Solutions Act. I think you know, age 688. And I know that you have some thoughts to share with us about that in representing the associate industries of Vermont. And I wanted to welcome you here and thank you for your patience and waiting for us to get off the floor. And if you wouldn't mind introducing yourself for the record as we record our hearing. Absolutely. So my name is William Driscoll with Associated Industries of Vermont. We represent primarily the manufacturing sector in the state. And certainly I appreciate the opportunity to talk to you today about this legislation and hope to continue to engage with you folks going forward. You know, I'd like to maybe address some sort of high level issues today and consider the follow-up to further comments in writing and more specifics. Obviously we engage in some of the questions and answers and discussions as you folks try to figure your way forward with this legislation. So I just want to maybe start by putting into context it's not that it shouldn't probably be pretty obvious where we're coming from but just the perspective we're trying to bring to some of the issues and specifics in the bill here. You know, for us this issue is not these issues because obviously this is a broad and complex topic. You know, it's not about whether or not to have climate policies our primary focus is how do we have the most reasonable and balanced policies. One of our concerns, not surprisingly, is trying to avoid undue economic harms but also harnessing positive economic opportunities as well as we, you know, get into eventually what are details and specific proposals. Because however important climate priorities are obviously there are many critical priorities that we're all charged with trying to address and although climate and economic priorities don't have to be in conflict and of an art they're always the possibility that they can be when you get down into specifics. So that's the perspective that I have some sort of basic comments or recommendations both on the climate council itself and sort of how it's set up and also the sort of implementation and accountability provisions further out on the bill. With regard to the council, again from the perspective of trying to make sure that you're getting the kind of perspectives that kind of balance input that I think is important both in putting forward things that you ought to do and things you ought to avoid. Obviously there are a lot of perspectives that the stakeholder groups represented already was proposed but I think it could be adjusted to include additional representation that I think is important to the topic. The first thing is if I recall correctly you have farm and forestry combined as one representative and I think honestly a lot of both the challenges and opportunities in farming and in forestry are not necessarily the same I think it would be beneficial to separate those two and have one from each. Second, having a representative from the manufacturing and industrial perspective I think would be helpful. Obviously a lot of emission concerns and economic concerns and opportunities stem from manufacturing and industrial activities so I think having somebody from that that sector would be an important ingredient and similarly transportation and freight providers you know from our perspective obviously our areas of either concern or opportunity are not just operations and facilities in Vermont but as you folks know fully well a big chunk of our emissions portfolio is transportation based and the hauling of supplies and goods primarily over the roads in Vermont is a big part of that issue but also very critical to our operations and so having folks that can speak to those issues who's important and that's not necessarily entirely within the manufacturing sector a lot of that activity operates on its own in terms of the business sector. So those would be the primary sort of membership recommendations. The other one which obviously you provide some flexibility in the subcommittee structure and approach but I think it would be helpful if in addition to I think you have two subcommittees specifically cited in the bill it would also be helpful to have one that's specifically charged with looking at avoiding or mitigating economic harm but also realizing economic benefits from again the interaction between whatever proposals come forward and you know in businesses in the state I think that's an important part of the overall discussion and I think having a subcommittee that makes sure that that's really looked at I think would be helpful as well. One question I have and this speaks to the sort of next area of suggestions is also the question of the decision making mechanisms of the council. Anybody who's ever put together, served on, or sat in the same room as study committees and things like that, that legislation push forward the whole question of do you have the right mix of people and do you have a fair decision making process is a very difficult one to get right and to figure out. And unless I missed it I don't think that this really speaks to exactly how the council will make its decisions like for example, you know, majority consensus unanimity, you know, that could be a factor in how the sort of the quality and the reception of their final product could affect that. I think that can be addressed with what we propose on the implementation side. So I want to say we have I think hopefully fairly straightforward but anybody has any questions about anything we can recommend so far? You are accurate, there's not in here a mechanism that says here's what the vote count is, for example. Right. If you know if it was that straightforward in terms of adoption, you're right. So what we would recommend the next part which obviously has to do with deadlines and schedules and frankly the cause of action and the role of the courts which we have concerns about that for a number of reasons. What we would really recommend is rather than what you have on the implementation side in the bill right now is you have the council make the recommendations and reports that they are required to do but have those not simply submitted to the legislature but require their approval by the legislature. That I think first of all given the balance of considerations that need to go into specific proposals I think having that go through the legislative process is both important and reasonable. That can also serve frankly to be a filter or if you will, for the other challenges of trying to make the council the perfect representative body with the best process for approval. I think similarly how we don't have study committees and working groups go straight from report to law or rule there's a good reason why those go through the legislative process I think given the significance of the issue and the policies and proposals that are likely to be coming forward it's all that much more important that those should be approved by the legislature. Just looking for clarification on what it is that should be approved so we require reports is that what should be approved or are you looking is it before that? So involved in the rulemaking? Right well so say the council's recommendations include the obvious one would be legislative changes but also specific rules that the agency would be required to put forward that the legislature would either adopt the legislative proposals or if there's specific rule direction and authority that's recommended that the legislature approves that. Similarly to how now the legislature passes laws that require rules for implementation. So it would give an opportunity for the legislature to review the actual specific proposals and approve them and if necessary maybe make adjustments if they see that there's good reason to do so. Again I think this addresses some questions whether in the collective judgment of the legislature whether the council has gotten everything right in terms of balancing priorities or specifics also addresses the question of you know getting that balance of representation and fairness and decision making correct as well. And I don't think that I don't see why having that legislative approval step would necessarily really slow the overall schedule or progress of what you're trying to achieve I don't think that I don't think the timing or the substance or the merits of what's done would change by requiring a legislative approval and I think you might be able to achieve some balance and judgment of that sort of a more sort of I want to say a hand-fisted in a clumsy way but in a more strict way that court intervention might be might lead to in terms of input and considerations that are brought to play. I think frankly sometimes especially if you look at you know court intervention where there's maybe some delay in the council's recommendations or delay in the rulemaking sometimes there should be a reason for those delays it's not necessarily something that warrants a court case just because you're off by a month or a couple months or whatnot there may be practical reasons for that and I think having the legislative approval process allows some more flexibility such a sensitive trigger on that and also when it comes to meeting the goals and having cases brought to the court to have the court direct rulemaking I think that raises questions in that kind of context are you going to have the same sort of public input process that you would have in the normal rulemaking process are you going to have a consideration of different perspectives and also some judgment in terms of weighing the process of specific proposals I think having the legislature really driving that allows for that much better than setting up a situation where where if you're off by a percentage point or two or more on those target deadlines I think it's a little more again it's the most more blunt instrument to use the courts and I think if you try to come up with guardrails or other ways to try to moderate that or what you're really doing is you're getting closer closer to replicating what might be a legislative process anyway so I'll stick with that on the outside so that's a very general area of recommendations there are reasons for specific things just before we move back to that one thing I'm hoping to think about more and come back more the bill makes reference to maintaining net zero emissions across sectors after 2050 and I think we would like to sort of consider more what that might mean in terms of say for example business activity or manufacturing activity that obviously has a greenhouse gas footprint regardless of how efficient you might be could be written into a situation where you could have a state-of-the-art facility but if you we actually have more manufacturing expansion here at Vermont are you going into situations where simply expanding activities that we might not want for various reasons might run counter to again a net zero concept part of that to be frank I need to talk to more people about what we say net zero can I just mention something because I think it's actually relevant to a number of your members and I think some of the consideration of this is reflected in looking at what New York has attempted to do and part of the New York law acknowledges that there are sectors of the economy that are going to be really challenged it's not clear what the technological path is to completely eliminate in 30 years fossil fuel as part of the fuel mix and greenhouse gas emissions as part of the manufacturing process that's specific to manufacturing and I think what New York has tried to do is provide a path of very strict and narrow for industries that have difficulty getting there finding where there is mitigation through other means that there may be industries they can't get off the fossil fuel may be able to reduce but not eliminate whether it's generally speaking through sequestration means or other can get to net zero even while they still may be emitting but it's a very prescribed and narrow path that was one of the concepts that was embedded in the New York legislation so that's something we can talk about more but I think it's very relevant to probably some of the people that you represent and certainly if we're talking about goals and aspirations trying to find ways forward I think that allows some flexibility as we get closer it's hard to predict what we're going to do today and whether if we're blocking ourselves into some things today that may not really work that there could be some challenges net actually I need to get back to another panel but one other one other aspect which again I'd like to come back to you folks I just want to sort of flag now on page 23 line 18 which is the only line reference I was going to make policy and currently the policy is based on a concept of lease cost which includes environmental priorities which I think would cover greenhouse gas reduction this bill would set greenhouse gas reduction outside of and alongside lease costs right now lease cost is a basket of critical priorities which are then balanced and by taking greenhouse gas reduction sort of out of the basket and having it co-equal with everything else combined again obviously employers can debate about what the lease really impacts but that raises a potential concern are we are we upsetting the balance of all the priorities that we need to focus and is that really appropriate or necessary and again I'd be happy to come back to you with more thoughts but just to look back and kind of skipped over this part as sort of following on to having the implementation authority and timing put into the hands of the legislature we would further recommend that that you remove the cause of action for the enforcement of it and frankly I don't think you need to make your goals mandatory if the legislature is going to be involved in implementing the targets along the way if for whatever reason the legislature takes actions that don't meet up with those mandatory bills I would presume it would be for some brief reason or not I think as long as you have certainly if you have an explicit cause of action and even if you simply make it mandatory without an explicit cause of action I think you just raise some potential legal concerns that I don't think are necessary but if you go along with the legislative approval as you go through the schedule the same schedule that council is already supposed to be working on there you are going to go back and track that up for you I think actually anything I could do to help I remember just a comment following up on what you said and what the congressman Chargers said about this a lot in terms of the manufacturing sector I'm very aware that depending on what it is the manufacturing and what the manufacturing process is that some industries are at this point in time and for the immediate for a simple future much more dependent on fossil fuels than others those that require huge amounts of heat as part of the manufacturing process that's where it comes from now and I read other alternatives coming along the pike but not next year 10 years maybe 15 years from now so I think that's something we should be cognizant of I think and it's just an observation because it came up with a particular suggestion or proposal in terms of representation on the council or anything like that at this point and I think a big part of that that part of it too in terms of what can be done and how it can be done the challenges like you say certain production activities lend themselves better or worse to electrification versus fossil fuel and frankly in terms of picturing a reduction in path the ability to convert from basically diesel to natural gas for example I know you folks may not be aware of this but access to natural gas and natural gas infrastructure is kind of a complicated issue so Heidi has experience on committees where I actually couldn't help but make everybody laugh quite a bit I come back more often hopefully I'll get you guys but yeah isn't this a complicated question and I'll just add as a footnote to Avram's point that something that we've had testimony on this committee and that other states have focused on it's not something that's in this bill is that the focus on emissions reduction should relate to what the contribution of that pollution is you know there are certain sectors transportation which you've highlighted but then also certainly thermal are the major contributors in Vermont and as you mentioned our manufacturers certainly engage in transportation and heat plant floors but the industrial sector as a contributor is a piece of the pie that's actually pretty modest in Vermont that's not to say that manufacturing businesses aren't meaningful contributors they are but in kind of looking at the pieces of the pie that we want to focus on we know where those are part of this is sort of the connection between manufacturing and some of the other sectors like transportation I guess one thing I would just point out there again not that folks don't see it this way too you can certainly look at it from the perspective of where are the contributions coming from but then the relative distribution of contributions is not necessarily the same as the availability of redresser measures in other words an area might be a big contributor but you might face some significant limitations on how you actually can go after that one diesel fuel transportation is a very good example that's certainly a chunk of the contribution but if you look at the efficiency of logistical planning and the technologies available there are big challenges in trying to reduce that contribution so the answer is an example of some of the things to look at airplanes might be an even better example and I'm sorry I don't mean to grab or keep a bad fit another question I think for folks and this gets to the rulemaking for A&R and the authority of it it wasn't clear to me looking at the bill obviously the council will come up with all sorts of recommendations but it looks like it's really A&R that's going to be doing the rules and so it raises the question of does does any recommendation of the council even if it's in an area that's not currently within the jurisdiction or the traditional responsibility of A&R some transportation sectors or other that is A&R actually going to be given both the authority and the responsibility of doing such rules as the bill is structured or as you might change it so that's just not a thing for us the quick answer is they would have the authority not necessarily the responsibility bringing people together in the council certainly allows regulation to occur there's nothing in the bill that precludes agencies that currently have authority to use that authority how the bill is currently drafted is at the end of the day A&R as the primary regulator of pollution air pollutant has the ultimate accountability for that but I think what the bill envisions is that different agencies in state government would collectively work to regulate Mike I want to make sure you get your question to go so thanks for coming here Bill I was listening but my computer wasn't catching up with the recommendations you had on the membership of the council so if you could go over that once again I'm very sure that farming and forestry should each have their own representative have that combined it should be represented from manufacturing slash industrial sector and also one from transportation slash freight manufacturing industrial sector and what was the other one transportation in our mind we're primarily thinking of goods and supplies commercial freight transportation but I think there are some recommendations out there that represent those things okay thank you any other questions great thank you thanks for joining us thanks for the specific suggestions yep


Thank you, Andres. First of all, thank you for inviting me. It's a great pleasure of me to be here. And today I will be talking about a topic that I've been working on lately in the last few years, mainly with Sebastiano Vina and sometimes with other people. So if you already know about centrality and the history of centrality, you may take a nap for the first part of my talk, but if you are not so familiar with the topic, then I think I will provide a sort of a bird-eye view on the subject and above all on the history of this topic. And then in the very last part I will focus on a specific measure of centrality, Page Rank. I have chosen Page Rank because this conference is about the Google metrics, so it's natural to take this as an example. And I will focus on a specific axiom that is rank monotonistic, but this will happen only in the last part of my talk. And if I will have enough time, I also want to show you a sketch of the proof. More or less the content of my talk will be as follows. First of all, I will start with a sort of a plea of centrality. I will try to convince you that centrality is very ubiquitous, very interesting, very simple, but very interesting, and it pops up in many different situations. And doing this I will mention the fact that by now we have literally hundreds of centrality indices. And of course I don't have time to present them all, but I will try to provide you a sort of a guide to try to classify. I will have a sort of a taxonomic approach in this talk, so I will try to allow you for a classification of centrality indices. And of course I will single out some of them. In this talk I will concentrate on six of them. Two for each of the three families that I will define. Then I will also try to convince you that a good way to study centrality measures is using axioms. This is not the only way, but it's a very nice one. And many people have tried or are trying to use this approach to study centrality, and by now we also have a jungle of axioms. So once more I will try to give you a guide and a taxonomy of the possible axioms. This is quite new. This is something we are working on these days. And I will focus on some important examples of axioms. And then in the very last part of the talk I will consider a rank monotonicity for page rank, and for this only case I will provide a sketch of a proof. Let me start with a very, very general scenario that I have in my mind and that explains why we as computer scientists come to like the idea of centrality. I described this as a, I said information retrieval system, but in fact it's a much more general thing. What I have in my mind is that I have a document repertoire, whatever it is. So it may be the set of pages retrieved by Google, or it may be the set of items that are on sale on Amazon, or it may be the members of a social network like Facebook. It can be many, many different things. I will call in each of these examples, I will call the items or the objects there, documents, and I write the D for the set of documents. You should think D is finite, but very, very large. And then there is a set of queries. Now the set of queries is typically infinite. You in most cases have a query language to write the queries, but don't think that the queries are always explicit. Is it? What should I do? So in some cases the queries are really explicit, right? When you type a query on a Google search engine, that thing is explicit. So it's an actual user typing an actual query using an actual query language. But in fact part of the query is implicit, because the query that arrives at Google is probably enriched with information that you don't explicitly type, like your location, whether you are typing on a mobile device or not, etc. So this is an example where the query is partly implicit and partly explicit. But in some cases the query is totally implicit. For example, in product recommendation on Amazon, say, then you are going to be recommended a product based on your action. There is no explicit query there. There is your last purchase or the fact that you have been looking for some specific thing on Amazon and so on. All of these I would call query. And then there is the system. So the system takes a query in and what it should do is selecting the appropriate documents, whatever they are. So for the search engine it should select the appropriate pages for your query, but for product recommendation it should select the product that you should be recommended, etc. And typically this is two things. One is a set of documents that are appropriate and besides for each of them a score. That is a value that measures how much that specific document is appropriate for your query or relevant for your query. And often, more often than not, D is endowed with a graph structure. The simplest example probably is pages retrieved by a search engine because in that case pages are part of a graph that is called a web graph. But in all the other cases you can think of the graph that is underlying D. Now the selection part of this can be in fact simplified. Many people prefer not to think of the selection, but rather to think that you score all the documents. Simply assign a very, very low number for the documents that you wouldn't select. With this simplification, which is not a big one, the whole system can be described simply as a function that for every query and document pair assigns a score. And the larger the score, the more appropriate is the document for the query. Now the second simplification that I will be doing is apparently more crazy because now I am assuming that there are no queries. This seems absurd, but in fact there are two good justifications for this simplification. The first one is that in some cases you are not interested in how relevant a document is for a query, but rather how important a document is overall. So you are interested in importance and not in relevance. The second justification for this simplification that I am going to apply is that in some cases you can take the query and modify the document repertoire so to reflect the query you have in your hands. And then after this you can throw away the query. In both cases the query is not there anymore and the system is now simplified as something that assigns a score to every document. The query is not there anymore. And the third simplification is that this scoring happens based only on the linkage structure. So we forget about the rest of information, only look at the graph structure and assign a score to every document. And this is what people call centrality. Centrality is a way to assign a measure of importance to every vertex in a graph. This is called centrality measure. Always think that the notes of our graph are the documents now. Keep in mind that this is called centrality index or centrality measure or centrality score. I prefer not to use the rank, the word rank if I can, because rank is something else. You use scores to assign a rank to the documents, right? You can sort the documents by decreasing score. This is what people do. But the rank is something different from the score. At the end of my talk this will be super clear. It's unfortunate that page rank is called page rank. The correct name should be page score because it assigns a score to every page, not a rank. But nonetheless, I will try to always refrain to using the word rank in this context. Now, where centrality was born, where the word centrality comes from, in fact the first usage of the word centrality comes from social sciences. And it was used for the first time by a group of sociologists working at MIT at the end of the 40s. This group was led by Alex Bavelas. And from this very seminal work there was a stream of works that started back then in the 50s and is still going on. Centrality is used a lot by social scientists. Psychometrists, sociologists, economists, they all use different kinds of indices for centrality. And here I wrote that it was brought to computer science through information retrieval. This happened sometimes with the advent of search engines. But this is sort of an up posterior reconstruction because as computer scientists we prefer to ignore the decades of history of centrality that preceded us. In fact, if you read, for example, the page rank paper, they make no mention of the fact that they are developing a centrality measure in fact. It is only recently that we are looking back to the past and see that many things that we as computer scientists are doing have been done much earlier, either by social scientists or by mathematicians. I will try to give you some pointers later. So as I said that it has a key role and by now there are so many centrality indices that it is sometimes difficult to have a guidance. Now the way I prefer to look at indices is by clustering them into three families. Remember that the overall purpose is to assign a score to every node in a graph that measures how important that node is. The first family is what I call path-based indices. Path-based indices are based on the number of paths or sometimes shortest paths passing through or arriving at the vertex you are interested in. Two examples in this family are very famous, they are called betweenness and cat's index. I will give you the exact definitions in a second. The second family that we like much more than the first one as computer scientist is the spectral family. It's a very general thing but we can say that it's based on some linear algebra construction or on some eigenvector that is defined on a matrix that is derived from the adjacency matrix of the graph. Page rank is a good example, you all know about it. Another example is a series index but there are many more.  Geometric indices are based on distances, distances from a vertex to the other vertices. Probably the best example is closeness, centrality, harmonic is another example. The six names of centrality indices that you see here will be defined in a while in my talk. I should say that these three families are not completely distinct and in fact the first two families have a large overlap, many times the same index can be defined using paths or using some metrics with a spectral construction. Let me give you the examples that I was mentioning. So I started with path-based centralities and as I said the path-based centrality depends on the paths entering or passing through a node and without further ado let me define first betweenness. Betweenness is very popular among social scientists. If you talk with people from social science they say that many of them think that it's the only measure of centrality. It was defined by Antonis in the early 70s and it goes like this. The betweenness of a node X is defined with this summation so you take every possible pair of vertices, YZ and you look at all the shortest paths connecting them and count how many of those shortest paths pass through X. You get that ratio there and you sum over all possible YZ. If you want it is proportional to the probability that if you select the shortest path that path passes through X. In my view this is not a measure of importance but it's rather a measure of robustness but still people use it many many times with the intention of measuring importance of nodes. Sometimes robustness is a measure of importance, it depends on your problem. Cat centrality was defined much earlier. It can be defined in many different ways but probably the simplest one is this one. First of all cat centrality is a parametric centrality measure so you have to decide a parameter that is called alpha in this formula. What you do is you count all the paths ending at X of length T for every possible T. Here I mean all the paths not elementary paths and this number is discounted exponentially by this factor here. So essentially a node is important if there are many short paths arriving at the node or many many long paths because long paths are counted less because there is this decaying factor. Starting from where? Wherever. They should end in X but they start whenever. You count all the paths. This is cat centrality. By the way many of these measures were defined originally on undirected graphs. I am in fact defining them for directed graphs and in these cases there is always the problem of whether considering outgoing paths or incoming paths. And I always use what is usually called the negative definition which means I consider incoming paths because in most situations you are more interested in incoming paths than outgoing paths and the importance is endogenous. It is given by people that is pointing to you. So these are the two definitions of the path-based tribe and let me move to spectrocentrality. Spectrocentrality in a way has a longer history and the first probably simplest definition of spectrocentrality is just take the left or right dominant eigenvector of the adjacent symmetric. This is called eigenvector centrality. It is not very used for many reasons but it is simple and sometimes it is considered. In fact we could trace back its usage from the works by Landau at the end of the 19th century. Landau was interested in evaluating the importance of chess players based on chess tournaments. He was manipulating matrices that contain 0, 1 and 1, half. 0 means a defeat, 1 means that x won over y and 1, half was used for a draw. Landau observes that if you take m times the all-one vector you get a reasonable measure because you are simply summing 1 for every win, 1, half for every draw and 0 for every defeat. But it is even better to consider the powers of m and at the end of a long discussion he says that what you actually need is solving this eigenvector equation. I think this was the first time, as far as we know, that somebody is trying to measure importance based on an eigenvector. This idea was brought to social sciences by Birch at the end of the 50s. I don't think he was aware of the work by Landau. I don't think he quoted Landau. Now, C. Lee, at the end of the 40s, proposed a similar idea to measure children's popularity. He had this group of children and he wanted to measure who was the most popular. He came out with this recursive formula, the popularity of x, of a guy x, is a summation of the popularity of the guy's y that like x. Here the arrow of the graph means y likes x. But there is a small variant with respect to eigenvector centrality that is the popularity of a guy is divided by the number of people he likes. This is the out-degree of y in the graph. This is the difference between eigenvector centrality and C. Lee's index. eigenvector centrality is the left dominant eigenvector of the adjacency matrix, whereas C. Lee's index is the left dominant eigenvector of g after raw normalization. I write g r for this. And there are two different indices. Let me come to page rank, which is something you all know. It was defined in 1998. The idea is that you can define it in many different ways. It's the left dominant eigenvector of this guy here. You recognize two things. You recognize that there is the raw normalized version of the matrix, like in C. Lee, but there is also this decaying factor. If you write this as a summation, you will see easily its similarity with cuts. In fact, cuts can be defined almost in the same way, except that in cuts you don't have the raw normalization. You can think that page rank is the son of two parents. One is C. Lee and one is cuts. Like in C. Lee, you got raw normalization, and like in cuts, you got the dampen factor. This is the recursive definition of page rank, but you all know about it. Or there should be a y here. I will use the definition later when I will talk about page rank. Remember that I'm using the slightly generalized definition with a preference vector v. In the original one in the paper by Brennan Page, v is the uniform vector. It doesn't change much. Let me pass to the last family, geometric centralities. Geometric centralities depend on distances. You can precisely define what I mean by using what I call the distance count function. The distance count function for the node x at distance t is the number of nodes z that are at distance t to x. If you let t range over all the possible positive integer, you get what I call the distance count vector. Of course, it's ultimately zero. After n, it will certainly be zero. This distance count vector is in one-to-one correspondence with something that is probably better known. It's called the neighborhood function. The neighborhood function is defined exactly in the same way, but with a less than or equal to here. Geometric centralities are the functions, the centralities, that are a function of the distance count vector. What I mean is that if two nodes have the same distance count vector, they will have the same centrality. For example, a trivially in-degree is geometric centrality because it's just the projection of this vector on its first component. The first component of this vector is the in-degree, which is by itself a measure of centrality, and it's the geometric centrality. Two more famous families of geometric centrality are closeness. Closeness is usually defined as the average distance from a node to all the other nodes in the network, but in fact, you want to take the reciprocal because you want important nodes to have large centrality as usual, so you take just the reciprocal of this summation here. You must be careful because closeness was defined by Bavellas. It was actually the first centrality measure that was defined in social sciences, but Bavellas had in his mind undirected and connected graphs. If the graph is directed and possibly non-strongly connected, you must be careful in this summation, and what you do is you take this summation only in the same connected component as x, or if you prefer, you take it only for the y's that are at finite distance to x. This causes a number of problems, a phenomenon that is called big in Japan effect. That is, the people that are living in a small community tend to be considered more important than they should. There are some ways to adjust this. One possible way is considering instead what is called harmonic centrality. The difference between the two formulas is very small, but at least here you can consider the summation over all possible y's with the proviso that when this guy here is infinite, you just sum zero. The idea of harmonic centrality was inspired by works by Markiore and Latoura of the early 2000s, but can be dated back to the 50s. So, with all these definitions in mind, remember that the six definitions that I gave are just the tip of the iceberg. There are so many possible centrality measures, but one problem that one often tries to solve is which centrality measure is better or how does this centrality measure compare to this other one? The study of centrality measures can be done either individually. You can focus on a specific measure and study it at depth, like PageRank and study how PageRank depends on the graph structure, on the parameters. The parameters in the case of PageRank are the damping vector and the preference vector, blah, blah, blah. So you can study a measure in depth individually or you can try to compare different measures and you can do both things in two different ways. One is by using some external source of ground truth if you have one. Or you can assume an axiomatic approach. You state a property that is desirable or undesirable for centrality and then you check if which of the measures do satisfy that property. I think that the axiomatic approach is nice because at the end of the day it allows you not to decide which measure is more important but rather how measures are related to each other. Much like the T axioms in topology that allow you to classify topological spaces. So axioms for centrality are not a new thing. You can trace the study of these axioms back to Sabidusi in the 60s. Many people are trying to do this. They are trying to define axioms that are suitable and then trying to see which measure satisfy them. Sometimes in a comparative way but sometimes aimed at specific indices. For example here I'm quoting two works where they only study axioms for pitch rank. Sometimes they even have like in this work they even have a sort of a hard core approach. They want to define a set of axioms that is necessary and sufficient for a given index. By the way they succeed in this case but only for a sort of degenerate version of pitch rank without alpha. Now once more I think that the axioms are so many that it is wise to try to classify them and I am trying to do that using three classes that are interesting. One I will call invariance properties then I will talk about score dominance properties then I will talk about rank dominance properties and I should say that there are also many other axioms that don't fall in these categories but the first three categories are interesting by themselves. Let me start precisely with invariance properties and first I want to give you a flavor of how those properties are usually stated. Usually they are stated like this. You have two graphs g and g prime and you have two notes one leaving in g and the other one leaving in g prime and these two things satisfy a number of constraints and under these constraints you want that the score of x in g is equal to the score of x prime in g prime. Now depending on the constraints the actual constraints you get different invariance properties. Let me start with a very simple one that is what I call invariance by isomorphism. This one is the following. g and g prime are two graphs that are isomorphic and f is the graph isomorphism. You take a node in g and it's a mage in g prime and you want the two notes to have the same score. Now this property is so obvious that in fact it is given for granted. When we say centrality we mean a measure that satisfies this action, right? Because we want centrality to depend only on the graph structure not on the name of the nodes. So when we define centrality we should always state it using this property which is fundamental even if usually people don't do that. So this is a very nice invariance property and all the centrality measures that I define and that are defined in the literature do satisfy this. You can do one step further and consider another kind of invariance which I call invariance by neighbors. Invariance by neighbors in this case you have just one graph g and two nodes living in the same graph and these two nodes have the same in neighbors and the same out neighbors. Under this condition you want x and x prime to have the same score. Otherwise said if you take two twin nodes that have exactly the same interface to the graph by interface I mean incoming edges and outgoing incoming arcs and outgoing arcs then you want them to have the same score. Is it satisfied by the measures I defined? Well yes, the answer is yes. In fact it is easy to verify that all geometric centralities do satisfy this property and the same happens for spectrocentralities and you can take this one step further, right? Because all the measures that I mentioned so far only depend on incoming links. So you may think that this property this invariant property can be strengthened by what I call invariance by in neighbors. Invariance by in neighbors simply means that if you have two nodes in a graph and they have the same in neighbors they should have the same score. Is it true for the measures I defined or isn't it? What do you guess? Well at the beginning when I saw this when I thought about these axioms I thought that all the measures that I defined in fact satisfied it but it's wrong, it's not true and I give you an example of why it's not true. I will show that geometric centralities in general cannot satisfy this kind of invariance. Take these two guys. So x and x prime are two guys living in the same graph they have exactly the same set of in neighbors but they may have different out neighbors. So the in neighbors are the same but the out neighbors can be different. Now take any other guy, z and consider a shortest path let's say from z to x. Now however this shortest path is done at the very last step it will contain an arc from one of the in neighbors of x to x and you can slightly modify this path to obtain a path which is exactly the same but going to x prime, right? So what I said is that a shortest path from z to x can be turned into a shortest path from z to x prime and the other way around. This means that the distance of the guys from x is the same as the distance of every guy to x prime. But there is this important difference I am considering a guy which is not x or x prime because x and x prime cannot be considered here the distance from x to x prime and the distance from x prime to x can be completely different and the result is that if you consider their distance count vectors they are almost the same but not exactly the same. There will be one single unit that is moving. For example here the single unit is moving from here to here because in this example the distance from x prime to x is 4 whereas the distance from x to x prime is 8 and since these two vectors are different in general you cannot assume that they have the same centrality and in fact we have counter examples for all the families of geometric centralities. This is a small difference but it's enough to make the axiom thing. On the other hand if you consider symmetric graphs this does not happen so they do satisfy invariance by enablers and by the way spectral centralities all satisfy invariance by enablers. So far I've considered invariance properties now I move to score dominance properties. Score dominance properties go like this you have two graphs, you have two notes some constraints and based on these constraints the centrality score of x and g is at least as large as the centrality score of x prime and g prime. Sometimes you require a strict larger, in this case I talk about strict dominance. One example of dominance that was considered recently in this wonderful work by Schock and Brandes of 2016 this by the way this is a very nice piece of mathematics they try to define a large family of centrality measures based on some semi-group construction but at some point they consider this dominance that I call dominance by enablers it goes like this you have two notes in a graph and the enablers of x are a subset of the enablers of x prime then you want the score of x to be not larger than the score of x prime. Why should it be like that? Well, because of course if x has a subset of it the enablers of x prime is that x prime is at least as important but we already know that in directed graphs for geometric centrality measures this property cannot be satisfied because if it would be satisfied also invariance by enablers would be satisfied In fact Schock and Brandes studied this property only on undirected networks. Another very famous example of score dominance is score monotonicity Score monotonicity goes like this pay attention because this is the property that I will be focusing on in the rest of my talk so you have a graph that does not contain a specific arc x to y you add this arc and what you want is that this addition will increase the score of y This is in fact the property that was considered by Katia in her talk two days ago she used to call it sensitivity So this property was studied a lot for many different indices and this is the result for the indices we are looking at You see that there is no apparent pattern There are some spectral measures like Silly that don't satisfy it PageRank does satisfy it and in a sense Katia gave it for granted during her talk If you add an arc to a node the score of that recipient node will always increase Betweenness does not satisfy it Cats does etc The situation is different on strongly connected graphs sometimes Let me just show you why on closeness and PageRank you have this situation First let me start with PageRank The score monotonicity for PageRank was proved back by Chen and others in this work of 2003 They in fact had to assume that all nodes have non-zero PageRank for their proof to work We generalized it under a much weaker assumption We only assumed that the source node has a positive score For closeness the fact that closeness in general does not satisfy score monotonicity can be shown on a very small example This is the example Take this graph here Now the centrality of Y is 1 There is only one node that can reach Y and it is at distance 1 But if you add the arc from X to Y now you have two guys at distance 1 and the overall centrality of Y becomes 1 half So in this example you add one arc to Y and you decrease the centrality of Y instead of increasing it On strongly connected graphs though closeness centrality does satisfy the axiom and the reason is very simple This is the definition of closeness In a strongly connected graph this summation ranges over all the possible Ys So after you add the arc you have exactly the same set of Ys over which the summation is done But adding an arc will in general decrease or not increase all the distances and there is at least one distance that gets decreased that is the distance X to Y So this summation strictly decreases which means that the closeness strictly increases Now score dominance properties are interesting but in fact they are not so interesting because as we saw in the talk of Maria two days ago it may be the case that you add this arc from X to Y and we know that Y increases its score but this doesn't tell us much because maybe there are other guys that increase their score as well and maybe they become more important than they used to be with respect to Y That's why people usually prefer to consider rank dominance properties Let me jump the general approach and go directly to the rank to the rank monotonicity definition that is the one that I want to focus on So this is the rank version of the score monotonicity property I discussed a second ago You have a graph, you add one arc X to Y and you want this to happen So if Y used to be better than Z then the addition of the arc the same must be true and if Y was at a tie with Z then after the addition either Y is still at a tie with Z or it is even better This is called rank dominance So you don't want only Y to increase its score but you want to keep its position in the rank And this is the weak version of rank monotonicity This situation requires that all ties are broken in favor of Y Not only you have the same position but you also break all the ties in favor of Y This table tells you what is the situation for rank monotonicity and by comparison here I am putting in parenthesis the score monotonicity properties You see that in all cases rank monotonicity was satisfied also rank monotonicity is satisfied and in many cases it is satisfied in a strict way A star here needs strict So at least for these six measures there is nothing surprising if the score increases also the rank is preserved and I will focus in the few minutes that I have I will focus on rank monotonicity for page rank The story goes like this In the work by Chen and others where they proved that page rank is score monotone they also proved the weak rank monotonicity under the assumption that every score is positive every page rank is positive We did the same more recently but we were able to prove the strict version of rank monotonicity with a much slightly stronger hypothesis that is we had to assume that the preference is everywhere positive This is stronger of course because if the preference vector is everywhere positive then all the scores are positive but we can prove that this condition is in fact necessary we have counter example of graphs that are strongly connected but our preference is zero and that fail to satisfy the strict version of rank monotonicity Now the difference between what Chen and others did in 2004 and what we did is also in the technique that we had to use because in their work if you look back at their work which was by the way very interesting they use the fact that the google matrix is a regular Markov chain Markov chain properties whereas we had to use properties of m matrices which we think are interesting because they are very general in fact what we did can be applied in many different way in too many different measures and we are able to apply our theorem at the same time to page rank and cuts for example so let me give you an idea of what we did we worked in a more general setting of which page rank is a special case and also cuts is a special case that is called dump spectral ranking you start with a non-negative matrix and with a factor that is more than the reciprocal of the spectral radius of m and with a strictly positive vector v and you define the rank to be like this now if you are confused by the generality of this definition think that you get page rank when m is the normalised version of g of the adjacency matrix and you get cuts when m is exactly the adjacency matrix but this is as you can see more general so this is the kind of centrality we are working on and the basic lemma that we use is the following it's an interesting lemma discuss it in detail because I think it can find applications even to other problems it's a lemma about inverses of m matrices so you have this m matrix here and you take its inverse and you define the rank by multiplying c by a preference vector v now consider in this matrix c two columns y and z and compute all the ratios between elements in the y column and elements in the z column of course to do this you must for the moment assume that z does not contain any zero now among all these ratios there is one that involves the diagonal in the column y this ratio here now it turns out that this ratio is the largest assuming that c y z to be non-zero this ratio is the largest of all ratios you don't need to write it as a ratio in fact you can write it this way so the only thing that you are assuming to be non-zero on this column is c y z this is the first property which is interesting and which we use and the second property as a consequence of this if you take the rank of y you well this guy can be multiplied by q and you get that r y is less than or equal to r z and as a consequence if r z is instead less than or equal to r y then q must be larger than one this is the property we this is the key property that we will use in the theorem so the theorem goes like this remember that we are looking at this guy here and 4-page rank m is the normalised version now let me observe that if you add an arc and look at the difference between the new matrix and the old matrix what you get is something like this all the rows that are not the x-row are zeros and the only difference is in the x-row and the x-row contains some negative entries that are the old out-neighbours of x and exactly one positive entry that is the in position y I call this vector delta and what we do is observe that the difference between the rank after the addition of the arc and the rank before the addition of the arc can be written in this way using Sherman-Morrison's formula kappa is a positive constant I will get rid of it in the next delta is the vector that I just showed you remember it is all zero or negative and it has exactly one positive entry and then there is this oh sorry and then there is this inverse of an m-metrics and this is where we use the lemma now what we need to prove is that if r y used to be better than r z now after the addition of the arc it is strictly better and in fact what we prove is that the increase in z the increase of score in z is less than the increase of score in y and the increase of score by this formula can be written in this way so what we need to prove is that the z-th element of this guy here is less than the z-th element of this guy here under this condition but the condition well you have to discuss many cases but the only non-trivial one is when the y z element of this matrix is positive the other cases are easy and in this case using the lemma by the fact that r z is less than or equal to r y we know that q is larger than or equal to 1 so if you look at this the y-th element of this guy here is delta times delta y times c y y c is this inverse matrix minus these elements these elements are all negative so I can write them as a minus and take in the absolute value and using the fact that q is larger than or equal to 1 we just collect q from here and get the inequality that we want in fact we prove the weak inequality now to go to the strict version of the inequality we need some more work but I wanted to give you the flavor of the proof because the proof is very different from the one that Chen and others used in their work they used the properties of regular market chains so what is the home message of my talk if there is one that I wanted to convince you and I hope that I did that centrality is important and it is very ubiquitous unfortunately we have so many centrality indices and I think we should make an effort to classify them in families our classification into three families is still very rough I think I think we can do a better job axiomatization is one of the ways in which I'm not claiming that it's the only one or that it's the best one but it's a good one and by now there are so many axioms that also axioms should be in a way taxonomized and this is the very beginning of an attempt to look for a taxonomy of axioms and be careful because sometimes you see a property that appears to be so trivial that is either true or false but sometimes proving that it is actually true or actually false requires a lot of work so in case you are interested in this kind of proofs you should be careful because it's much wider than you think at the first site thanks a lot for your attention thank you very much questions you have examples of our security examples of what examples with concrete data with Google yes so I didn't want to present this in this talk but the axiomatic approach is not the only one that we use in fact it's not the first one at the beginning we looked at actual data and the well web graph we used the main example coming from the web and the comparison between all the indices I don't have here this slide but you can see that some indices behave are pretty much alike like for example page rank is very very similar in practice to in degree page rank is also quite similar to cuts but it's for example very different from closeness and by the way it is based on these observations that we observe that betweenness is an index by itself it doesn't have any relation with any other index so I mean in a nutshell we did and other people did a lot of work on practical graphs and on these indices the problem is that what you come out with is you get these two ranks and then what you have closeness centrality of the old UK pages of a large web graph and then you have page rank centrality you can compare them with the Kendall style let's say to see if they are similar or not to decide that they are not similar then deciding which one is better is a tough job you must have some ground truth to do that you discussed the change of page rank when you add a link so if you have weighted graph and you change the weight can everything conventionally analyze them probably so yes, most probably so so it's a matter of sensitivity every result we used is in fact continuous so I think it's easy to generalize to weighted page rank but in that case it is not really clear of what you want for rank monotonicity that if you increase by epsilon the weight I discussed the variation by epsilon of weight I am pretty sure that for every positive epsilon if you increase the weight by epsilon you get strict dominance but I'm not 100% sure but probably it's like that one question you spoke about centrality in terms of vertices vertices but are there measures of centrality for edges betweenness was originally defined for edges yes of course many people study centrality on arcs or edges depending whether you are considering directed or undirected networks which is also quite interesting you can do it directly or you can do it indirectly looking at how sensitive the network is by the deletion of an edge an edge is important when you delete it you change a lot you disrupt a lot the centrality of the nodes actually I have two questions one is you mentioned you have counter examples if I understand when the damping vector has zeros and then that monotonicity doesn't work does it have an effect on the usability of page rank or is it some degenerate things only it's only the strict version that doesn't hold I don't think it has any practical impact because the weak version will in any case hold as long as all pages have positive rank the other question is are there axioms for stability like adding something will not significantly change the course well you can think of rank monotonicity as a form of stability what you are saying is that the addition of that arc does not change the relation between the target node and the other nodes I'm not sure about the rest of the network I'm not sure whether if you add an arc here you cannot change the relation between two guys that are not involved in the addition probably so this would be another very interesting axiom to study thank you