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If you turn on the TV, or read the financial press, financial pundits & investors alike are talking about “The Everything Bubble.” There are multiple stages of financial bubbles, & while each mania has its own nuances, in retrospect there are almost always commonalities that the market & market participants experience.

## Howard Marks: “We Are In An Everything Bubble”

Dr. Jean-Paul Rodrigue published a widely circulated economic bubble chart in which he identified four distinct phases of a bubble, referred to as the (i) Stealth Phase (ii) Awareness Phase (iii) Mania Phase & (iv) Blow-Off Phase.

Looking at drawdowns in the S&P 500 as a proxy for “bubbles,” according to BofA we’ve had 17 different 30% pullbacks in the S&P 500 since it was created, with 4 occurring in the last 30-years (2020, 2007, 2002, 2000) where the “Blow-off” phase was catalyzed by the COVID-19 pandemic, Global Financial Crisis (GFC), the aftermath of 9/11 & the Tech Bubble.

What’s been less discussed is the mental psyche of market participants across each stage, particularly as the awareness / mania phase extends longer and longer. This is where we find ourselves today. Despite the 30% drawdown in February to March 2020, the snapback was so quick, that for most market participants it has felt like an extension of the bull market that risk assets have been in since March of 2009. Some market participants are referring to this as “The Everything Bubble.”

There are seemingly two distinct phases that we’ve transitioned between in the current “mania phase” that has been exacerbated by the unprecedented fiscal and monetary stimulus, real yields, and corresponding risk asset appreciation.

• There Is No Alternative (“TINA”) 1.0 - As signs of mania / blow-off phases begin to percolate, the early movers looked to go to cash as it was “safe.” We saw personal savings at all-time highs, corporates raised cash & suspended dividends / buybacks / M&A, etc….
• TINA 2.0- After months of negative real-yields, the mentality has started to transition to the detriments of holding cash, as the negative wealth effect has been very real, watching asset prices continue to appreciate around them, moving out the risk curve, which typically culminates in the blow-off top.

The “Everything Bubble” is overly simplistic & not fair to all parts of the market so we’re looking to disaggregate what’s causing these “bubbles” and where opportunities may be if we do see the blow-off phase.

# Monetary / Fiscal Stimulus

Since the COVID-19 pandemic began, there’s been \$32T of fiscal and monetary stimulus, which is the largest stimulus as a percentage of global GDP that the world has ever seen. Global Central Banks have spent \$834M per hour buying bonds over that time and now ~25% of all government bonds are negative yielding, with a significantly higher percentage at negative real-yields, given inflation.

Source: Bank of America

In the US alone the Fed has bought \$4 trillion of bonds in the past 18 months, while the US government is spending \$875M per hour in 2021.

This is driving people further out the risk curve, and as we’ve said in the past, “QE Infinity turned your savings account into your checking account, the bond market into your savings account, the equity market into the bond market, the venture market into the equity market, while given rise to the crypto market as the new venture market.”

## Real Asset Response

We’ve seen a significant appreciation in commodities with the S&P GSCI +120% from the lows last March, and commodities such as Copper +102%, Aluminium, +82%, Corn +80%, Nickel +49%, Gold +22%, etc… We saw Lumber +300% at one point (and now it’s still +20% from March ‘20 levels), while the US CPI Used Car Index is +45% YoY.

If we look at the S&P CoreLogic Case-Shiller 20-City Composite City Home Price Index, the YoY move is approaching higher levels than we saw in 2004-2006 & now the median house price as a multiple of median income is at an all-time high.

##### Real Asset Bubble?

Are we in a real asset bubble? It depends

Commodities: As we think about the macro tailwinds around electric vehicles, policy driven decarbonization targets, and renewable energy, one of the common themes is the “electrification of everything.” There is a “Big 6” story going on in the commodity world right now between Copper, Nickel, Silver, Lithium, Cobalt, and Aluminum out of which the supply / demand dynamic is most clear for the first three. Take copper for instance, which thus far has never seen demand destruction & to-date, there are no known technologies that are able to conduct electricity other than Copper. In autos, high efficiency motors consume 75% more copper than a standard motor, EV’s consume up to 4x copper in terms of weight compared to ICE’s, and require copper-intensive charging stations. Further, renewable energy including wind & solar consume 4-5x the amount of copper compared to traditional fossil fuel generated power. So there’s a real demand driver for copper.

Compare that to something like when Lumber was up 300% in under 12 months due to a shortage that was shortly alleviated, there are pockets of froth but for most commodities the market does a good job of offsetting increased demand, by increasing supply, when feasible.

Used Cars: Used car price increases are unsustainable. Due to supply chain shortages and one-time behavioral shifts, car manufacturers haven’t been able to produce as many cars needed to fulfill demand. This is alleviating given the average ownership tenure of a car, and the easing of supply chain bottlenecks. That 2006 Ford Mustang probably won’t continue to appreciate at 20%+/yr.

Homes: Probably. The only time we’ve seen house prices go up this fast was the precursor to the housing crisis. That said we saw a material shift for the better part of a decade post the GFC, leaving the US structurally short homes, which the National Association of Realtors (NAR) pegs at 5.5M units (2M single-family homes, 1.1M units in buildings with 2-4 units, and 2.4M units in buildings with 5+ units), while Freddie Mac estimates the current shortage at 3.8M units.

From 1968 to 2000 there was an annual average of 1.5M new units, from 2010-2020 that’s been hovering around ~1.0M. But as home builders observe what’s going on with housing prices, coupled with millennials finally buying homes, you can expect supply to start to creep upwards towards that 1.5M and likely 1.8-2.0M / year which should put a cap on how high existing home prices can continue to appreciate, and even result in “down rounds” as supply once again catches up to demand.

Equity Market Response:

How have equity markets responded to this unprecedented level of fiscal & monetary stimulus? In the spirit of TINA 2.0, funds have flown into equity markets at unprecedented rates.

As of early August, Global Equity funds have seen \$605B of inflows YTD, or \$1.015T anualized, equating to \$3.98B/day. Global Equity funds had seen \$727B of cumulative inflows over the last 25 years (‘96-‘20); that’s \$115M/day. 2021 is on pace to be 40% higher than the prior 25 years combined. While we’ve seen an increase in the number of companies coming public via IPO’s / SPAC’s the number of investable companies hasn’t kept pace with the degree of inflows, resulting in re-ratings. Looking at the total market cap of stocks with P/S in excess of 20x we’ve surpassed the tech bubble high by nearly ~\$1.0T.

If it feels like the market is hitting all time highs every other day, that’s because it is. We’ve had 50 all-time highs for the S&P 500 as of 8/24 year-to-date, only twice in history have we seen this many through August; 1964 & 1995.

US Equity Market Bubble?

While bears point to all time high P/E, CAPE Ratio, etc… that fails to take into consideration the equity risk premium & where yields are. During the ‘99-‘00 period the 10-year was 5-6% vs ~1.0-1.5% today.

Looking at the S&P 500 today versus on this date (August 24th) in 1999 (arbitrary- yes) we are trading at a cheaper P/E multiple on estimated ‘21E / ‘22E EPS than where the market traded on realized earnings in ‘99-‘00, but notably the ERP was negative then as you earned a higher yield on treasuries than earnings yield on the market, whereas today that’s +3.2%-3.6%.

There are arguments to be made that the broader “fair market multiple” has re-rated higher from a mid to high teens P/E multiple that is sustainable due to several factors:

• Composition Changes- The largest market constituents today are growing faster, more profitable, with higher margins than the largest companies of ~20-50 years ago, warranting a higher multiple all-else equal.
• Ease of Diversification- The creation of ETF’s, and the ease of which investors can invest in 500 or 2,000 different companies with as little as \$5-\$10 can support higher multiples than a highly concentrated portfolio.
• Global Nature- The percentage of ex-US sales for large US listed companies continues to increase offering enhanced geographic diversification than ~20-50 years ago.
• Rates- As countries increase debt, they can’t afford to have real rates higher as interest would become the greatest single expenditure, resulting in a higher equity multiple all else equal.

“This time it’s different,” are always famous last words, but besides being the same “index” there’s very little in common with the S&P 500 of today and that of 30-40 years ago.

Post GFC if you look at the S&P 500 market cap divided by the Fed Balance sheet it’s almost been a flat line; “Don’t Fight the Fed.” From January 2009 until today that ratio has averaged 4.6x with a low of 3.4 (March ‘09), and a high of 6.7 (Dec ‘19); today it sits at 4.7x right in-line with that decade + average.

In a TINA world, US equities don’t scream “bubble” at current levels.

##### Private Equity / Venture Reponse?

The fundraising environment has been robust; whether its’ KKR or BX raising \$59B / \$37B during 2Q, or the \$350B of capital that’s flown into APO, APRES, BX, CG, & KKR over the trailing twelve months; there’s been an insatiable appetite for allocators to put money to work in private equity. Global PE dry powder hit an all-time high of \$1.9T in January of 2021, & once those funds have capital, they need to deploy it, to repeat the cycle. Buyout deal activity in 1Q & 2Q21 set quarterly records, with \$354.2B total in 1H21 across 993 deals, already surpassing 2020’s total of \$257.3B across 1,429 deals. Given the public market valuations, capital from SPAC’s, and the public market receptivity to growth more broadly, the average buyout multiple hit 11.4x EBITDA in ‘20.

##### Venture:

We’ve seen similar activity across the venture ecosystem on the capital raise side, with firms like Index Ventures raising \$3.1B a year after a \$2.0B raise, Accel raising \$3.0B, A16z raising a \$2.2B crypto dedicated fund, etc…. GP’s have raised \$74.1B year-to-date, not far from the record of \$81.0B raised during the totality of 2020.

According to the Pitchbook / NVCA Q2 2021 Venture Monitor, we saw a quarterly record in 1Q21 with \$75B of venture dollars deployed, & 2Q matched that record with another \$75B deployed, leading to \$150B in 1H21. Mega-rounds (\$100M+) in ‘21 have already reached an annual record high of \$85.5B. We’ve seen increased participation from “non-traditional’ VC investors, such as mutual funds, hedge funds, corporate investors, & crossover investors. “An estimated 3,301 deals have received investment from a nontraditional institution (81.8% of the 2020 record high), representing \$115.9 billion in deal value (nearly matching 2020’s total). While deal count participation rates for these institutions have only gently ticked upward in recent years, the deal value represented by nontraditionals has skyrocketed. The median late-stage deal size with nontraditional investor participation eclipsed \$43 million in 2021, nearly \$18 million higher than the same median in 2020."

H1 has seen an explosion in crossover investor participation, totaling \$63.5B of capital across 524 VC deals, on track to surpass \$100B by year-end.

Given the amount of capital we’ve seen an increase in valuations across both early & late-stage deals. Per Pitchbook, “On the valuations front, the median and average early-stage pre-money valuation in H1 2021 has also drastically expanded to \$42.0 million and \$96.1 million, respectively—a notable jump over 2020’s \$30.0 million and \$61.7 million.”

Looking at FinTech specifically, we’ve seen a record amount of capital investments, larger deal sizes, and a notable spike in median / average valuations.

Paul Tudor Jones wrote an excellent letter last May entitled, The Great Monetary Inflation, where he outlined assets to own as inflation hedges in which he identified Gold, The Yield Curve, NASDAQ 100, and Bitcoin as the assets most likely to outperform.

“I also made the case for owning Bitcoin, the quintessence of scarcity premium. It is literally the only large tradeable asset in the world that has a known fixed maximum supply. By its design, the total quantity of Bitcoins (including those not yet mined) cannot exceed 21 million. Approximately 18.5 million Bitcoins have already been mined, leaving about 10% remaining. On May 12th Bitcoin’s mining reward — the pace at which the supply of Bitcoin is increased — will for the third time be “halved” (falling from 12.5 to 6.25 Bitcoins per block of transactions added to the blockchain). Future halvings will likewise occur approximately every four years consistent with Bitcoin’s design, thus continuing to slow the rate of supply increase and causing some to estimate that the last available Bitcoin will not be mined for another 100+ years. This brilliant feature of Bitcoin was designed by the anonymous creator of Bitcoin to protect its integrity by making it increasingly near and dear, a concept alien to the current thinking of central banks and governments."

This turned out to be largely the right call, particularly on the Bitcoin front.

#### Crypto Bubble?

Now that the “market cap” of crypto is +1358% over the past 18 months & BTC is +885%,

are we in a crypto bubble? We think it’s important to bifurcate between large cap crypto vs. some of the more speculative fringe, while diving into BTC, Layer 1 Smart Contract Protocols / DeFi, and NFT’s.

#### Bitcoin

On Black Thursday of last year when Bitcoin sold off as much as ~50% intraday few would’ve predicted 18 months later we’d be sitting at a price of \$50,000 with corporates such as Square, Tesla, and Microstrategy buying Bitcoin on their balance sheet, insurance companies like MassMutual buying \$100M, wirehouse banks offering Bitcoin access to their PWM clients, or that a \$1.2T infrastructure bill (crazy in its own right) would be held up as crypto lobbyists had their first taste of the Big Show, and the SEC would be showing body language that a futures-based BTC ETF might be approved (finally) in 4Q21. As it sits here at a \$900B “market cap,” BTC is behind the USD, Euro, CNY, and Yen as the fifth most “valuable” currency in the world. Through this lens you can argue that’s a stretch given the limited amount of commerce that is conducted via BTC. But the bull thesis for some time has been “digital gold” and that’s what the corporates / pensions / insurance companies / macro funds are all buying into, so fiat’s probably not the right comparison.

While we don’t have a precise estimate of the market cap of gold at current prices, a safe range is somewhere between \$9.0-\$13.0T which would correspond to a BTC range of \$428K-\$620K per BTC or up 775%-1163% from current levels. There are very few macro bets where an investor could deploy billions of dollars and have 8.5-12.5x upside with likely 60-70% downside.

Could Bitcoin go to zero and really that’s down 1 versus up 8.5-12.5x skewing risk / reward slightly? Sure. But Stanley Druckenmiller first bought Bitcoin because of something Paul Tudor Jones told him,

“Do you know that when Bitcoin went from \$17,000 to \$3000 that 86% of the people that owned it at \$17,000, never sold it?” Well, this was huge in my mind. So here’s something w/ a finite supply & 86% of the owners are religious zealots." There’s a lot more “downside support” then the most vocal bears would lead you to believe because bitcoin is “backed by nothing” (despite the largest dedicated resources of compute power in the world). What makes something a good store of value? PTJ ranks assets across four characteristics: • Purchasing Power- How does this asset retain its value over time? • Trustworthiness- How is it perceived through time and universally as a store of value? • Liquidity- How quickly can the asset be monetized into a transactional currency? • Portability- Can you geographically move this asset if you had to for an unforeseen reason? Bitcoin scores higher than gold on liquidity / portability today, fails on “trustworthiness” due to gold’s multiple thousand year head start, but purchasing power since BTC was created skews in BTC’s favor. Below is a chart of BTC / Gold ratio. Bitcoin bears will often point to MySpace / Facebook, or Blackberry | Nokia versus Apple, to highlight that the “first” technology doesn’t always win. That shows a fundamental misunderstanding of Bitcoin’s purpose as a store of value. You can’t replicate Bitcon’s origin story with an anonymous founder, who would be amongst the richest people in the world and ~12 years in still hasn’t sold any BTC, a fair launch with no VC’s, where anyone in the world had the ability to participate, done completely open-sourced with volunteers on a global basis, dedicated to see the “project” through, with 0 FTE, and 0 marketing budget. Subsequent attempts have been funded by venture funds, early crypto whales, with central points of failure, and known teams, which really violates that “sovereign-grade censorship resistant” property needed to compete with gold. It’s not technology that will win this, it’s security of the network & decentralization, where Bitcoin has what looks to be an insurmountable lead. As we see a generational wealth transition from Baby Boomers, to MIllennials, Gen Z’s, and Gen Alphas, that grew up natively digital, and spend all their time in the metaverse; what store of value would they prefer? As Travis Kiling puts it a “a non-sovereign, hardcapped supply, global, immutable, decentralized, digital store of value” that serves as an insurance policy against monetary & fiscal policy irresponsbility from central banks & global governments, or a yellow rock? At a ~10% probability of a BTC / gold flippening, and the fact that you have a perpetual call option without an expiry, or theta decay, makes BTC a really compelling macro asset and at current levels hard to consider a “bubble,” even if we see another 50-75% retracement (as we already did this year). #### ETH / Other L1’s There’s been a lot of debate regarding DeFi over the past 18 months, as the total value locked (TVL) in DeFi smart contracts now sits at \$80.5B up from \$925M last March. We wrote about DeFi and the evolution of existing financial market infrastructure here. ETH continues to have dominant market share in DeFi, Web 3.0, NFT, and gaming, as the leading L1 protocol, as measured by market cap, developer activity, TVL, and notional dollar value of transactions. Is ETH in a bubble? Raoul Pal recently dubbed ETH potentially the “greatest trade” in the history of financial markets over the next six to nine months. He mentioned that ETH was set up for the “triple halving” with EIP 1559 dramatically reducing the supply of ETH by burning fees & thus burning ETH and lower fees, & the launch of ETH 2.0 with the transition to proof of stake (“PoS”). He highlights that the number of ETH addresses is growing faster than BTC’s (60% per year vs. 30% per year), and more importantly that the transaction count is growing as well. In early August ~26% of all ETH was locked into DeFi smart contracts, leaving just ~13% of ETH left on exchanges to trade. In his view, this is the perfect storm of a massive supply shock for an asset that’s exhibiting exponential demand. We can argue that ETH faces much stiffer competition from other L1 competitors, and this is truly a battle on technology. To win DeFi, Web 3.0, NFT’s, & gaming you need to have a high performant blockchain with the appropriate trade-offs between the Blockchain Trilemma of Scalability, Security & Decentralization. Right now, ETH is working on scalability as the network can handle ~30 TPS, with block times of 14 seconds, and transaction fees range from a few dollars to a few thousand dollars depending upon congestion of the network. To onboard billions of people into Web 3.0 that’s not sufficient. Users will expect experiences comparable to that of using the internet today, where things are instant / free, with an intuitive UI/UX, and we’re definitely not there yet. We’re not believers that there will be “one chain to rule the world,” but liquidity & therefore network effects tend to coalesce around a handful of purpose built winners for technology (e.g., AWS / Azure / Google) and trading (e.g., ICE / NDAQ / CME / CBOE) so there’s a need to put a stake in the ground. Similar to Bitcoin’s origin, ETH has benefits in that it created generational wealth for thousands, if not tens of thousands of developers who participated in the ICO, or bought ETH early on, giving them a very loyal developer ecosystem that’s incentivized to see it work. While decentralized purists will cringe when they hear this, just like when investing in companies, when investing in protocols it’s an investment on the team. Vitalik Buterin is the most well-known developer in crypto outside of Satoshi Nakomoto and for good reason. A bet on another protocol is a bet on someone not named Vitalik. The prize for the winner of L1 protocols is likely \$1.0T+, so at \$375B is ETH in a bubble? If ETH 2.0 fails, sharding proves to be suboptimal, the decentralization actually gets in the way of product road map / progress, then unequivocally yes; \$375B would be the 24th largest company in the world. What if they’re able to execute? There remains asymmetric upside, without having to take a binary bet due to liquidity. Underwriting ETH from current levels is either a trade in the near-term on supply / demand dynamics, or over the next 3-5 years is an explicit bet that they will execute on their stated roadmap (which has been in the works since 2016), and be one of the first & last protocols standing as it pertains to L1 value capture.

Who else has a chance as it pertains to L1 value capture? Enter Solana. There’s been a lot of talk on twitter about “Solana Summer” and for good reason as the price has increased from \$24 to \$70 from Memorial Day to present (August 24th) , and that’s up from under \$2.00 to start the year. Packy McCormick just published a great overview of Solana that speaks for itself, but as we see it Solana is the most notable direct competitor to ETH today. Solana block times are 0.4 seconds, transaction fees are a fraction of a penny, and max capacity is currently ~65,000 TPS. You can monitor the state of the network here. Ironically, failing to gain significant venture interest in 2017-2018 when protocols such as Dfinity, Polkadot, Tezos, Cosmos, EOS, Algorand, Dfinitiyt, etc… raised billions of dollars, was the best thing that could have happened to them. Similar to ETH, they had to ship product and build a community of developers organically. They launched their first hackathon in November of ‘20 with 1,000 registrations, their second was in February with 3,000 and by May their third had 13,000. You’re seeing real teams build on top of SOL and real transaction activity. The FTX team built Serum on top of SOL, with SBF dubbing it “the most performant blockchain.” We’ve now seen ~25B transactions on SOL since the mainnet launched in March ‘20 (this includes consensus messages, so the real number is probably ~½ of that) compared to ~1.25B for ETH and ~665M for BTC. When you offer users nearly instant & free transactions, more of them occur. To be clear this is sometimes a vanity metric that earlier L1’s tried to manipulate, but utilizing your Phantom wallet, sell something on Serum for USDC, transfer it to Saber, stake it, and compare that to any other crypto experience. Is SOL in a bubble? Given the prize to win L1, the adoption by both traditional trading firms and crypto native firms, and a growing and organic developer ecosystem, coupled with the discount to other L1’s, ETH, you can make an argument that it’s still undervalued here on a “relative value” basis, if it’s able to be one of the L1 winners. What about other large cap cryptocurrencies? When you look at assets like Cardano “valued” at “\$90.5B” (without smart contract functionality ~4+ years in), XRP at “\$54.6B,” and DOGE at “\$39.1B,” it’s tough to see a world where tokenholders generate above market returns holding those assets over the next 3-5 years. It’s crypto and anything could happen, but those most certainly represent pockets of bubbles amongst “large cap crypto.”

As you move further down the list, there are hundreds of “zombie chains” with multiple hundred million dollar + valuations, countless chains where valuation is only ascribed to “circulating’ supply, and zero use cases outside of speculation. For some, that’s all you need, but over time attention drifts, and most of this is definitely signs of froth and excess in the crypto market and we expect to see significant “wealth” destruction there over the next 3-5 years.

#### NFTs

NFT’s have clearly made it mainstream (and had what most viewed as the ultimate contra-indicator) when there was an SNL skit in late March. While some early excess was removed from the market, since that time, we’ve seen some mind numbing NFT sales with Beeples \$69M sale at Christie’s, the price of the “cheapest” Cryptopunk increasing 15x+ since the beginning of the year, Visa entering the mix with a \$150K purchase re-rating the market overnight, one punk selling for a record \$7.8M, with others listed “offered” at \$5.5-\$7.7M, Degen Apes selling out in 8 minutes despite servers crashing, and Justin Sun spending \$500K for a pet rock.

Any of those things in a vacuum would seem like a “top” indicator, just as a video clip of LeBron dunking in a meaningless regular season game selling for \$208K (which would be a top 4 card for him), seemed to suggest that in February for NBA Top Shot (it was to the day), or when Cryptokitties saw several 6 figure sales in Dec ‘17, also marking the top. Similar to collectables / art in the physical world, something is worth what someone else is willing to pay for it, and there will be no difference in the world of NFT’s. Owning a cryptopunk means you can afford a cryptopunk. There’s debate over whether or not this is the new wealth flaunt, such as owning a Ferrari or a Patek Phillippe, in the world of the metaverse. Are we in an NFT Bubble? There are undoubtedly pockets of excess and just as the world of high end art, sports memorabilia, cars and watches have created generally accepted “valuation” rankings based on cultural significance, scarcity, longevity, etc… the world of NFT’s will coalesce around a similar hierarchy of value, but it’s just not there yet. Will people be able to sell “pet rocks” for \$500K in 3 years? Probably not. Will the buyer of a random LeBron dunk in a meaningless regular season game be able to sell that for \$200K? Highly unlikely. Will Visa be able to sell it’s “floor punk” for$150K in 3-5 years? That’s more questionable. Right now people are having fun, and as Fred Wilson put it, NFT’s sit at the nexus of “gaming, online communities, and social networks.”

We’re in the very early innings of crypto’s first real “consumer opportunity” and when you see Jay-Z put a cryptopunk as his twitter profile picture, or Paris Hilton receiving a Degenape, or the explosion of play-to-earn with Axie Infinity (so much so that it caught the eyes of the Philippine regulators highlighting the need for players ot pay income tax on the game), it’s hard to ignore the momentum of potential consumer experiences built on top of the crypto stack and right now NFT’s are showing the way.

Most of this isn’t for us, but neither is art, cars, watches, or wine. Just be careful of the market’s unrelenting supply creation machine to substantiate demand; as it’s much easier to create digital supply than it is in a physical world.

#### Conclusion

If global central banks and governments are going to continue to print money, investors are faced with a TINA 2.0 predicament, where cash is literally burning a hole in their pockets, pushing them not just into risk assets, but further out the risk curve, exacerbating wealth inequality along the way, leading to even further risk taking.

So are we in an everything bubble? There’s most certainly pockets of excess in nearly every corner of the financial markets, but there’s also ample opportunity.


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Open Access 01.12.2020 | Research

# Network-based indices of individual and collective advising impacts in mathematics

verfasst von: Alexander Semenov, Alexander Veremyev, Alexander Nikolaev, Eduardo L. Pasiliao, Vladimir Boginski

Erschienen in: Computational Social Networks | Ausgabe 1/2020

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## Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Introduction

In recent years, universities and other research institutions have put a lot of emphasis on assessing and enhancing the productivity of their faculty. One aspect that has been traditionally deemed important in these efforts is the number and quality of a researcher’s publications. The popular metrics of publication productivity include various quantities based on an individual’s citation record (e.g., total number of citations, weighted citations, i10-index, h-index, etc.), typically accounting for the “prestige” measures of publication outlets (e.g., journal impact factors, 5-year impact factors, SNIP, CiteScore, etc.). However, besides publication output, another—possibly equally important—aspect of the academic profession success is associated with advising and mentoring Ph.D. students. One can argue that a successful academician is not only the one who publishes many highly cited articles, but also the one who successfully advises students, and further, whose students in turn become successful academic advisors, thus ensuring the continuity and prosperity of an academic discipline. Indeed, in the modern era, many universities emphasize the importance of effective mentorship and post-graduation academic productivity of their Ph.D. students.
This paper makes contributions towards a systematic network-based analysis of large-scale Ph.D. student advising data. We define and interpret a family of new network-based metrics (collectively referred to as “a-indices”) that can be used for “ranking academic advisors” using the academic genealogical records of scientists. We rely on the well-known web-based Mathematics Genealogy project resource that has collected a vast amount of data on Ph.D. student advising records in mathematics-related fields.
Due to its popularity and public availability, MathGenealogy dataset has been used as a testbed in several previous studies. The basic characteristics of the MathGenealogy network snapshot from 2011, as well as those of the underlying network of countries, were presented in [1]. In [2], the authors analyzed the performance of students of those individuals who were near the beginning versus near the end of their academic careers and revealed interesting insights. Another study [3] used the data of Ph.D. degrees granted after 1973 and used it to compose a network of universities, where some of the universities were then labeled as strong sources (“authorities”) of Ph.D. production, while the others were labeled as strong destinations (“hubs”). The authors of [4] presented a comprehensive analysis of the MathGenealogy network with respect to the classification of mathematics-related subjects, as well as most influential countries in terms of the Ph.D. graduates output. Further, they revealed the major “families” of mathematicians that originated in certain root nodes (“fathers” of mathematics’ genealogical families), in the different “eras”, covered by the project data. A new concept of eigenvector-based centrality was defined and tested on the MathGenealogy network in [5]. In [6], the authors proposed the so-called “genealogical index” for measuring individuals’ advising records. As it will be seen below, one of the indices proposed in this paper can be viewed as a special case of the “genealogical index” proposed in [6].
This paper takes a further step towards studying and ranking academic advising impact using MathGenealogy social network. The emphasis of this study is on taking into account not only the number of students advised by an individual but also subsequent academic advising records of those students, while providing the respective metrics that are easy to calculate, understand, and interpret. It should also be noted that this study does not aim to explicitly compare the proposed indices with other metrics/results available in the aforementioned related literature. However, we believe that the presented approaches and results provide a new perspective on this interesting subject and further demonstrate the utility of social network analysis tools in the considered context.
The paper is organized as follows. In the next section, we briefly describe the MathGenealogy dataset and provide its basic characteristics along with definitions and notations that will be used in the paper. Next, we define and interpret the family of “a-indices” that we propose for ranking academic advisors. We then extend these definitions to take into account co-advising. Finally, we present the results obtained on the most recent snapshot of the MathGenealogy dataset, as well as investigate the evolution of individual and collective a-indices over the past several decades.

## Data description, notations, and basic characteristics of MathGenealogy network

To facilitate further discussion, we first describe the MathGenealogy dataset and provide its basic characteristics, as well as define graph-theoretic concepts that will be used in the paper.

### Data description

The data were collected from the Mathematics Genealogy Project website1 using a web-crawler software. The dataset contains the records about nearly 231,000 mathematicians (as of July 2018). The information for each mathematician in the database includes name, graduation year, university, country, Ph.D. thesis topic and its subject classification, as well as the list of students advised by this individual. This available data allowed us to construct the directed network of advisor–advisee relationships.
Due to the fact that the considered dataset is a directed network, it is represented by a directed acyclic graph $$G=(N,\mathcal {A})$$, with a set of n nodes, N = $$\left\{ 1,\ldots , n\right\}$$, and a set of m arcs (links) $$\mathcal {A}$$, where the mathematicians are represented by the nodes of the graph, and the relation “i is an advisor of j” is represented by an arc from i to j. The in-degree ($$\text{deg}^{\text{in}}(i)$$) and out-degree ($$\text{deg}^{\text{out}}(i)$$) of node i are the numbers of the arcs coming into and going out of node i, respectively. Clearly, the in-degree of node i is the number of this individual’s Ph.D. dissertation advisors (equal to one for many nodes in the network, although a substantial fraction of nodes do have higher in-degrees), whereas the out-degree of node i is the number of Ph.D. students that this individual has successfully graduated. Node j is said to be reachable from node i if there exists a directed path from i to j. The number of links in the shortest path from i to j is referred to as the distance between these nodes and denoted by d(ij) ($$d(i,j)=+\infty$$ if there is no such path). A group of nodes is said to form a weakly connected component if any two nodes in this group are connected via a path and no other nodes are connected to the group nodes, where the directions of arcs in a path are ignored.
The harmonic centrality of node i is defined as $$C_h(i) = \sum _{j \in N} \frac{1}{d(i,j)}$$ [7, 8]. The decay centrality of node i is defined as $$C_d(i) = \sum _{j \in N} \delta ^{d(i,j)}$$ [9, 10], where the parameter $$\delta \in (0,1)$$ is user-defined, although it is often set at $$\delta =1/2$$, which is the value used in this study (it is assumed that $$1/d(i,j)=\delta ^{d(i,j)}=0$$ if $$d(i,j)=+\infty$$).

### Basic characteristics of MathGenealogy network

The retrieved network had 12,263 weakly connected components, with the giant weakly connected component having 208,526 nodes and 238,212 arcs (thus containing about 90% of all the nodes in the network). All the computational results presented below were obtained for this giant component. Further in the text, we will use the term “network” implying this giant weakly connected component.
The analysis of many basic characteristics of an earlier snapshot of this network was conducted in [1]. Since such analysis is not the main focus of this study, we report only some of these basic characteristics for the most recent snapshot that are relevant to the material presented in this paper. The distribution of out-degrees in this network is presented in Fig. 1. As one can observe, it does resemble a power law, although it is not a “pure” power law, which is consistent with observations for many other real-world networks [11].
The out-degree correlation for all “tail-head” (or, “advisor–student”) pairs of nodes corresponding to all arcs (directed links) in the considered directed network was calculated as follows. Consider an ordered list of all directed links $$l \in \{1, \ldots , |\mathcal {A}|\}$$ in the network, let i and j be the head and tail nodes of link l, and let $$\text{deg}_l^\text{{out}}(i)$$ and $$\text{deg}_l^{\text{out}}(j)$$ be their out-degrees, respectively. Thus, we have an array of size $$|\mathcal {A}|$$ of head nodes (denote the average out-degree of all nodes in this array by $$\overline{\rm{deg}^{\rm{out}}(i)}$$) and an array of size $$|\mathcal {A}|$$ of tail nodes (denote the average out-degree of all nodes in this array by $$\overline{\rm{deg}^{\rm{out}}(i)}$$). Then, the out-degree correlation (also sometimes referred to as the out-assortativity) can be calculated as:
\begin{aligned} r_{out} = \frac{\sum _{l=1}^{|\mathcal {A}|}({\text{deg}}_l^{\text{out}}(i) - \overline{{\text{deg}}^{\text{out}}(i)})({\text{deg}}_l^{\text{out}}(j) - \overline{{\text{deg}}^{\text{out}}(j)})}{\sqrt{\sum _{l=1}^{|\mathcal {A}|}({\text{deg}}_l^{\text{out}}(i) - \overline{{\text{deg}}^{\text{out}}(i)})^2}\sqrt{\sum _{l=1}^{|\mathcal {A}|}({\text{deg}}_l^{\text{out}}(j) - \overline{{\text{deg}}^{\text{out}}(j)})^2}} \end{aligned}
The value of the out-degree correlation for this network was found to be approximately 0.055. This implies that on average there is a very minor correlation between the mentorship productivity of an advisor and a student. Therefore, we believe that in the proposed metrics and rankings of academic advisors it makes sense to “reward” those prolific advisors whose students are also successful academic mentors.
As for the in-degree distribution, it is not surprising that the majority of the nodes have in-degree equal to one. However, the network contains over 30,000 nodes with in-degree greater than one, which means that a substantial fraction (about 15%) of the mathematicians in the dataset had more than one Ph.D. advisor. Therefore, it is important to take into account the effects of co-advising, which is why we define “adjusted” versions of the proposed metrics (indices).

## Advising impact metrics

In this section, we define four metrics (“a-indices”) that we believe are appropriate for quantifying an individual’s advising impact, with a focus on taking into account the mentoring success of an individual’s students (going beyond just the number of the Ph.D. students that an individual has graduated). One way to address this is to consider the numbers of students and students-of-students, whereas another approach is to take into account all the academic descendants of an individual. These considerations are reflected in the following definitions.
Definition 1
(a-index) The a-index2 of an individual i is the largest integer number n such that the individual i has advised n students (Ph.D. graduates) each of whom has advised at least n of their own students (Ph.D. graduates). Equivalently, this is the largest number n of out-neighbors of node i in the directed network such that each of these neighbors has out-degree of at least n.
Definition 2
($$a_\infty$$-index) The $$a_\infty$$-index of an individual i is the total number of their academic descendants, computed as the largest number of distinct nodes that are reachable from node i through a directed path.
Definition 3
($$a_1$$-index) The $$a_1$$-index of an individual i is the harmonic centrality of the corresponding node i in the directed network: $$a_1 (i) = C_h(i) = \sum _{j \in N} \frac{1}{d(i,j)}$$.
Definition 4
($$a_2$$-index) The $$a_2$$-index of an individual i is the decay centrality (with $$\delta = \frac{1}{2}$$) of the corresponding node i in the directed network: $$a_2 (i) = C_d(i) = \sum _{j \in N} \frac{1}{2^{d(i,j)}}$$.
It can be seen from Definitions 14 that the a-index is a measure of the most “immediate” advising impact of an individual, which takes into account their advising success simultaneously with the advising success of their students.3 Note that the a-index is similar to the h-index well-accepted for citations record evaluation; however, it turns out that it is rather hard to achieve a double-digit value of the a-index over one’s academic career due to the fact that graduating a Ph.D. student is generally a less frequent event than publishing a paper. As it can be seen in Table 1, the highest a-index value in the considered dataset is 12 (achieved by only four mathematicians). Note that a relevant study [6] reported only one mathematician with the value of a-index ($$g_{(1)}$$ measure in their terminology) equal to 12. Overall, the a-index may be applicable as a metric of the advising impact for middle- to late-career academic scientists.
Table 1
Top individuals by a-index, with the a-index of at least 10 and their corresponding adjusted a-index
a-index
Name
Country of Ph.D.
12
Heinz Hopf
1925
Germany
11
12
Jacques-Louis Lions
1954
France
11
12
Mark Aleksandrovich Krasnoselskii
1948
Ukraine
12
12
Erhard Schmidt
1905
Germany
10
11
Andrei Nikolayevich Kolmogorov
1925
Russia
10
11
C. Felix (Christian) Klein
1868
Germany
10
11
1923
Germany
9
11
Karl Theodor Wilhelm Weierstrass
1841
Germany
9
11
John Torrence Tate, Jr.
1950
United States
11
11
Ernst Eduard Kummer
1831
Germany
10
11
Reinhold Baer
1927
Germany
8
11
Salomon Bochner
1921
Germany
11
11
David Hilbert
1885
Germany
10
10
Lothar Collatz
1935
Germany
9
10
Günter Hotz
1958
Germany
10
10
Pavel Sergeevich Aleksandrov
1927
Russia
10
10
Edmund Hlawka
1938
Austria
9
10
Phillip Augustus Griffiths
1962
United States
9
10
Michael Francis Atiyah
1955
United Kingdom
9
10
Haim Brezis
1972
France
10
10
Thomas Kailath
1961
United States
10
10
R. L. (Robert Lee) Moore
1905
United States
10
10
Alan Victor Oppenheim
1964
United States
10
10
Shiing-Shen Chern
1936
Germany
10
10
Elias M. Stein
1955
United States
10
10
Richard Courant
1910
Germany
9
10
Hellmuth Kneser
1921
Germany
9
10
Emil Artin
1921
Germany
10
10
Lipman Bers
1938
Czech Republic
9
10
Issai Schur
1901
Germany
8
10
Roger Meyer Temam
1967
France
9
10
John Wilder Tukey
1939
United States
9
10
Philip Hall
1926
United Kingdom
10
10
Beno Eckmann
1942
Switzerland
9
10
Oscar Ascher Zariski
1925
Italy
10
Note that the a-index can be extended in a straightforward fashion to reflect a more “long-term” advising impact of an individual by considering third, fourth, etc., generations of an individual’s students as it was proposed in the definition of the “genealogical index” in [6]. However, the main issue with this approach is that close to 100% of the mathematicians in the considered dataset would have zero values of such index, which would not allow one to effectively rank advisors’ long-term impacts using this metric.
Therefore, in order to provide more practically usable quantifications of “long-term” advising impacts of individuals, especially for those scientists who are in the late stages of their careers and for those who have lived and worked centuries ago, we propose the $$a_1$$, $$a_2$$, and $$a_\infty$$ indices. The $$a_\infty$$-index essentially assigns equal weights to all the academic descendants of an individual, whereas the $$a_1$$ and $$a_2$$ indices prioritize (with different weights) the immediate (directly connected) students and students-of-students while still giving an individual some credit for more distant descendants. Possible practical interpretations of these indices are as follows.
The $$a_\infty$$-index is appropriate for ranking the “root nodes” of the mathematics genealogy network, that is, nodes with zero in-degrees, which essentially correspond to “fathers” of mathematics’ “genealogical families”, such as those described in [4]. It is not practically significant to calculate this index for nodes with non-zero in-degree values, since their predecessors in the network would obviously have higher values of this index. Thus, the $$a_\infty$$ index is interesting primarily from the perspective of history of mathematics, although it can certainly be calculated very easily for any contemporary mathematician.
On the other hand, the $$a_1$$-index and $$a_2$$-index do not necessarily possess the aforementioned property of the $$a_\infty$$-index: the values of these indices may be higher for contemporary mathematicians than for the “fathers” of genealogical families due to the fact that an individual’s immediate students and any other early-generation students attain higher index values than do any distant descendants. These indices are based on the well-known concepts of harmonic and decay centralities, which makes them easy to calculate and interpret, and hence, attractive from a practical perspective. These indices can be applied to an academic advisor from any era, thus providing a universal tool of assessing the academic advising impact. However, it is still likely that the advisors in the late stages of their careers would have higher values of these indices (especially the $$a_1$$-index that gives higher weights to distant descendants) than those in early-to-mid-stages of their careers. This is not surprising, since these indices are designed to assess the long-term advising impact beyond the number of immediate students.
Further, note that there are several natural extensions of these definitions. First, all of these indices can be adjusted by taking into account the effects of co-advising, that is, giving a special treatment to the cases when multiple individuals have advised the same student j (that is, with node j having multiple incoming links). These particular extensions are addressed in greater detail in the next section. Second, the a-index can also be defined for a specific country or university (similarly to the h-index of a journal among citations metrics), that is, considering the respective country or university as a “super-node”, with the outgoing links directed to all the Ph.D. graduates ever produced (or produced during a specific time frame) by this country or university, respectively. The resulting collective advising impact values for universities and countries, based on MathGenealogy dataset, will also be presented below.

In this section, we define the extensions of our basic indices (Definitions 14) to handle the cases of co-advising, that is, the situations where one Ph.D. student was co-advised by more than one individual. It makes practical sense to introduce these definitions due to the fact that a substantial fraction of the individuals in the considered dataset were advised by more than one advisor. The basic assumption that we make in the definitions below is that the credit for advising such a student is split equally between each of the co-advisors (i.e., if there are n listed co-advisors for a student, then each of the co-advisors receives 1 / n credit for graduating the student).

### Adjusted $$a_\infty$$, $$a_1$$, $$a_2$$ indices

The definitions of $$a_\infty$$, $$a_1$$, $$a_2$$ indices can be modified to take into account co-advising as follows.
Definition 5
(adjusted $$a_\infty$$-index) The adjusted $$a_\infty$$-index of an individual i is the total number of their academic descendants weighted by the reciprocals of their in-degrees, that is, $$a_{\infty , adj}(i) = \sum _{j \in N} \frac{1}{deg^{in}(j)}\mathbb {1}_{\{d(i,j)< +\infty \}}$$, where $$\mathbb {1}_{\{d(i,j)< +\infty \}}$$ is the indicator function corresponding to the condition that node j is reachable from node i through a directed path.
Definition 6
(adjusted $$a_1$$-index) The adjusted $$a_1$$-index of an individual i is defined as $$a_{1, adj} (i) = \sum _{j \in N} \frac{1}{ deg^{in}(j)}\frac{1}{d(i,j)}$$.
Definition 7
(adjusted $$a_2$$-index) The adjusted $$a_2$$-index of an individual i is defined as $$a_{2,adj} (i) = \sum _{j \in N} \frac{1}{ deg^{in}(j)} \frac{1}{2^{d(i,j)}}$$.
As one can clearly see from these definitions, the values of these adjusted indices are always less than or equal to the respective values of their “regular” counterparts, as common sense would suggest.

The above definition of a-index can also be modified to take into account co-advising, although this extension is not as straightforward as those in the previous subsection. The “adjusted a-index” of node i can be calculated as follows:
1.
Calculate the “adjusted” out-degree of node i: $$deg_{adj}^{out}(i) = \sum _{j: (i,j) \in \mathcal {A}} \frac{1}{deg^{in}(j)}$$. Clearly, this value can be fractional and is reduced to simply the out-degree of node i if none of the students of the corresponding individual i were co-advised.

2.
Compute and sort the adjusted out-degrees (defined as indicated above) of all nodes $$\{ j:(i,j) \in \mathcal {A} \}$$ in the non-increasing order. Denote this sorted array as $$D_1, D_2, \ldots$$ and let $$D_k$$ be the kth element of this array such that k is the largest integer satisfying $$\lceil D_k \rceil \ge k$$. Calculate $$\min \{ D_k, k\}$$.

3.
Calculate the adjusted a-index of node i, $$a_{adj}(i)$$, as the minimum over the values obtained in the steps 1 and 2 above.

This computational procedure ensures that the adjusted a-index of any node i is always less than or equal to its “regular” a-index, whereas the possibility of fractional values of the adjusted a-index provides a more diverse set of its possible values. This would potentially allow one to create a more “diversified” ranking of academic advisors based on their own productivity and productivity of their students, while taking into account co-advising.

## Results for MathGenealogy dataset

In this section, we present the results obtained on the MathGenealogy network using the metrics proposed above. Figure 2 shows the distribution of the values of the a-index and the adjusted a-index over the entire network. One can observe that while the “regular” a-index is always integer by definition, the adjusted a-index does often take fractional values, especially for lower spectrum values of the index, thus providing a more diverse set of possible values in a ranking. Further, Table 1 provides a ranking of top academic advisors with an a-index of at least 10, many of whom are prominent mathematicians from the nineteenth and twentieth centuries (note that none of the mathematicians who worked before the nineteenth century made it into this ranking). Their respective adjusted a-index values are also given in the same table for comparison. One can observe that this ranking would change if it was done using the adjusted a-index, thus showing that co-advising is indeed a significant factor to consider in this context.
Table 2 presents the collective advising impact rankings of universities and countries based on their respective values of a-index. It can be observed that universities and countries with prominent reputation in mathematics-related research fields lead these rankings, which shows that (i) not surprisingly, there is correlation between collective university-scale and country-scale research and advising impacts, and (ii) the a-index appears to be a realistic and appropriate metric for collective advising impact of a university or a country. Note that we do not consider adjusted a-index in this case (although it would be possible), since it is rare in the dataset that an individual’s co-advisors come from different universities or countries.
Table 2
Top universities and countries by a-index
University name
a-index
Country
a-index
Harvard University
31
United States
54
Princeton University
30
Germany
45
University of California, Berkeley
29
United Kingdom
33
Massachusetts Institute of Technology
28
Russia
31
Stanford University
28
Netherlands
29
The University of Chicago
25
France
26
Lomonosov Moscow State University
25
Switzerland
25
University of Cambridge
24
Austria
22
Columbia University
24
21
ETH Zürich
24
Belgium
19
Georg-August-Universität Göttingen
22
India
19
22
Sweden
18
California Institute of Technology
22
Ukraine
17
University of Michigan
21
Australia
17
University of Oxford
21
Romania
17
Universiteit van Amsterdam
21
Poland
17
Yale University
20
Spain
17
University of Illinois at Urbana-Champaign
20
Israel
17
Universität Berlin
20
Japan
16
Ludwig-Maximilians-Universität München
20
Italy
15
Carnegie Mellon University
20
Finland
15
Figure 3 shows the distribution of regular and adjusted $$a_1$$ and $$a_2$$ indices in the network. It appears that both of these distributions are close to power-law, whereas the range of values of the $$a_1$$-index is larger than that of the $$a_2$$-index, which follows from the respective definitions. Tables 3 and 4 present the rankings of the top 25 advisors by regular versus adjusted $$a_1$$ and $$a_2$$ indices. For each index, mostly the same group of advisors appears in the regular versus adjusted index rankings, although their order slightly changes in both tables. Moreover, one can observe that the $$a_1$$-index-based ranking favors earlier generations of mathematicians (those from sixteenth, seventeenth, and eighteenth centuries), whereas the $$a_2$$-index-based ranking features mathematicians from the nineteenth and the twentieth centuries. This is a direct consequence of the impact of the different weights given by these indices to distant academic descendants of an individual.
Table 3
Top 25 individuals ranked by the $$a_1$$-index (left) and adjusted $$a_1$$-index (right)
Name
Year
$$a_1$$-index
Name
Year
Adj. $$a_1$$-index
Simeon Denis Poisson
1800
11800.58
Simeon Denis Poisson
1800
10486.00
Abraham Gotthelf Kästner
1739
10719.19
Abraham Gotthelf Kästner
1739
9509.77
Joseph Louis Lagrange

10557.30
Joseph Louis Lagrange

9380.81
Pierre-Simon Laplace

10555.30
Pierre-Simon Laplace

9379.31
Jakob Thomasius
1643
10254.40
Jakob Thomasius
1643
9175.63
Leonhard Euler
1726
9969.036
Emmanuel Stupanus
1613
8852.32
Emmanuel Stupanus
1613
9907.44
Leonhard Euler
1726
8836.15
Christian August Hausen
1713
9712.28
Christian August Hausen
1713
8621.04
Johann Friedrich Pfaff
1786
9601.81
Friedrich Leibniz
1622
8565.26
Friedrich Leibniz
1622
9569.92
Giovanni Beccaria

8491.40
Giovanni Beccaria

9556.55
Jean Le Rond d’Alembert

8491.15
Jean Le Rond d’Alembert

9555.55
Johann Friedrich Pfaff
1786
8479.49
Carl Friedrich Gauss
1799
9395.80
Carl Friedrich Gauss
1799
8264.47
C. Felix (Christian) Klein
1868
9316.05
Petrus Ryff
1584
8262.28
Petrus Ryff
1584
9245.31
C. Felix (Christian) Klein
1868
8111.17
Johann Bernoulli
1690
9126.35
Johann Bernoulli
1690
8090.71
Johann Andreas Planer
1686
8885.37
Johann Andreas Planer
1686
7889.29
J. C. Wichmannshausen
1685
8882.86
J. C. Wichmannshausen
1685
7887.46
Johann Elert Bode

8707.06
Felix Plater
1557
7748.45
Felix Plater
1557
8669.28
Johann Elert Bode

7693.68
Jacob Bernoulli
1676
8400.77
Jacob Bernoulli
1676
7447.01
Nikolaus Eglinger
1660
8398.77
Nikolaus Eglinger
1660
7445.01
Julius Plücker
1823
8210.41
Johannes W. von Andernach
1527
7322.90
Johann Pasch
1683
8189.74
Guillaume Rondelet

7296.20
Rudolf Jakob Camerarius
1684
8189.74
Otto Mencke
1665
7274.00
Table 4
Top 25 individuals ranked by the $$a_2$$-index (left) and adjusted $$a_2$$-index (right)
Name
Year
$$a_2$$-index
Name
Year
Adj. $$a_2$$-index
David Hilbert
1885
1099.72
David Hilbert
1885
949.74
C. Felix Klein
1868
1016.04
C. Felix Klein
1868
873.30
C. L. Ferdinand Lindemann
1873
907.25
C. L. Ferdinand Lindemann
1873
780.80
Erhard Schmidt
1905
667.56
E. H. Moore
1885
597.00
E. H. Moore
1885
639.77
Erhard Schmidt
1905
550.32
Ernst Eduard Kummer
1831
636.78
Ernst Eduard Kummer
1831
535.59
K.T.W. Weierstrass
1841
575.10
K.T.W. Weierstrass
1841
484.32
Julius Plucker
1823
522.36
Solomon Lefschetz
1911
466.55
Solomon Lefschetz
1911
510.06
Julius Plucker
1823
449.19
R. O. S. Lipschitz
1853
508.52
R. O. S. Lipschitz
1853
436.90
Oswald Veblen
1903
474.34
Oswald Veblen
1903
431.94
Richard Courant
1910
458.12
Richard Courant
1910
400.25
Heinz Hopf
1925
446.18
George David Birkhoff
1907
388.12
George David Birkhoff
1907
415.53
Heinz Hopf
1925
349.33
Jacques-Louis Lions
1954
385.92
Nikolai Nikolayevich Luzin
1915
335.95
Nikolai Nikolayevich Luzin
1915
366.49
Jacques-Louis Lions
1954
329.44
Simeon Denis Poisson
1800
362.25
A. N. Kolmogorov
1925
326.89
A. N. Kolmogorov
1925
361.89
Simeon Denis Poisson
1800
320.97
Ferdinand Georg Frobenius
1870
354.50
Gaston Darboux
1866
311.96
Gaston Darboux
1866
346.64
Michel Chasles
1814
309.62
Michel Chasles
1814
337.36
H. A. Newton
1850
301.44
G. P. L. Dirichlet
1827
335.53
Ferdinand Georg Frobenius
1870
294.73
Ludwig Bieberbach
1910
334.29
C. Emile Picard
1877
287.67
Edmund Landau
1899
330.46
G. P. L. Dirichlet
1827
283.56
H. A. Newton
1850
323.17
Edmund Landau
1899
280.94
The ranking of individuals with in-degree zero in the network (that is, “fathers” of genealogical families) by their $$a_\infty$$ and adjusted $$a_\infty$$-index values is given in Table 5. The top-ranked scientist with respect to both of these indices is Sharaf al-Din al-Tusi, who lived in the twelfth century and currently has 149,942 academic descendants.
Table 5
Top 25 individuals with zero in-degrees ranked by the $$a_{\infty }$$-index (left) and adjusted $$a_{\infty }$$-index (right)
Name
Year
$$a_{\infty }$$-index
Name
Year
Adj. $$a_{\infty }$$-index
Sharaf al-Din al-Tusi

149942
Sharaf al-Din al-Tusi

134493.04
Elissaeus Judaeus

149909
Elissaeus Judaeus

134464.20
Jan Standonck
1474
149852
Jan Standonck
1474
134420.70
Cristoforo Landino

149821
Cristoforo Landino

134398.29
G. G. M. Groote

149820
Moses Perez

134395.79
F. F. R. Radewyns

149820
G. G. M. Groote

134393.87
Moses Perez

149818
F. F. R. Radewyns

134393.87
Ulrich Zasius
1501
149817
Ulrich Zasius
1501
134392.37
G. Hermonymus

149807
G. Hermonymus

134385.84
Jean Tagault

149750
Francois Dubois
1516
134341.45
J. ben Jehiel Loans

149750
Jean Tagault

134341.45
Francois Dubois
1516
149750
J. ben Jehiel Loans

134340.62
Johannes Stoffler
1476
149719
Johannes Stoffler
1476
134316.79
Nicole Oresme

147113
Nicole Oresme

131926.62
Luca Pacioli

147109
Luca Pacioli

131923.12
Bonifazius Erasmi
1509
147108
Bonifazius Erasmi
1509
131923.12
L. von Dobschutz
1489
147108
L. von Dobschutz
1489
131922.62
Johann Hoffmann

147073
Johann Hoffmann

131897.42
Friedrich Leibniz
1622
146244
Friedrich Leibniz
1622
131125.67
Thomas Cranmer
1515
142171
Thomas Cranmer
1515
127225.92
Ludolph van Ceulen

137372
Ludolph van Ceulen

122765.09
Marin Mersenne
1611
127076
Marin Mersenne
1611
113204.87
Paolo da Venezia

125964
Paolo da Venezia

112062.20
Sigismondo Polcastro

125961
Sigismondo Polcastro

112059.20

125959

112058.70

## Evolution of individual and collective a-indices in MathGenealogy dataset

As a natural further step in the analysis of individual and collective advising impacts using a-indices, we consider the dynamics of year-by-year evolution of the aforementioned indices over the past several decades. Specifically, we consider the time period starting from 1900 till 2017 (which was the last full year for which MathGenealogy data was collected in this study). The main reasons for considering only the data starting from 1900 are that (i) the growth of mathematics as a major research field occurred during the twentieth century with many new Ph.D. degrees awarded during that time frame, and (ii) the collected data itself is more reliable and complete for this most recent time frame, which makes the results on a-indices evolution corresponding to this time interval more interesting. We should also note that some of the plots presented below reflect the data starting from 1950, which is done for visual clarity purposes.
Figure 4 shows the year-by-year evolution of a-index values of top 10 mathematicians (according to their a-index value at present, as indicated in Table 1) starting from 1900 and until 2017. An interesting observation is that for most of these individuals, it took around 20–30 years to grow their a-index from 0 to 1 (that is the time period from the year an individual received his/her own Ph.D. degree to the year when his/her first student successfully graduates a student of their own). Further, it took another $$\sim$$30 years to grow their respective a-index value from 1 to around 10. The overall time period of 50–60 years to grow the a-index from zero to a high value of 10 or more is on the same order of magnitude as the length of a lifetime academic career (i.e., from the receipt of a Ph.D. degree till retirement). This shows that most of these “high-impact” advisors followed a similar temporal pattern of their careers. This observation is also consistent with the intent for this index to reflect an individual’s career-long rather than short-term advising impact.
It should also be noted that the “outliers” in this plot are E.E. Kummer and K.T.W. Weierstrass, who received their own Ph.D. degrees substantially earlier than the other individuals in this list, and their a-indices were already equal to 6 in 1900. Interestingly, both of their a-index values have “saturated” at 11 around 1920 and have not changed since then. This is most likely due to the fact that all of their direct descendants (students) finished their own academic careers by that time; therefore, they did not produce any more students after that, which means that the respective a-index cannot increase anymore. Thus, the a-index is a good measure of an individual’s lifetime advising impact; however, it does not reflect any further advising impact that an individual might achieve after the end of his/her own and his/her students’ academic careers. On the other hand, an individual’s $$a_1$$, $$a_2$$, and $$a_{\infty }$$ indices clearly can grow indefinitely, even decades or centuries after the end of one’s career (as it will be illustrated below). Therefore, a long-term advising impact may need to be evaluated by considering a combination of metrics (such as the indices defined in this paper) rather than by taking into account only one metric.
It is also worth mentioning that a collective a-index of a university or a country does not exhibit the “saturation” behavior that was mentioned above for an individual a-index. Indeed, a university or a country would typically keep producing Ph.D. graduates indefinitely (unless a university/country ceases to exist). Figures 5 and 6 illustrate the evolution of collective a-index values corresponding to top universities and countries (according to their current a-index values as shown in Table 2). For visual clarity, these plots are shown starting from 1950 rather than 1900.
For universities’ collective a-index values, there were several lead changes during 1900–1950 (not pictured), with Princeton being top-ranked for most of the 1950s and 1960s (briefly overtaken by the University of Chicago in mid-1950s), whereas in 1968 Harvard took the top-ranked position, which it has held till now. It should be also noted that Stanford has made a big jump from number 10 to number 4 in the a-index ranking during the past half-century. As for countries’ a-indices evolution, the United States passed Germany as number one in the collective a-index ranking in 1956 and has held this top position since then.
Note that the collective a-indices of most universities and countries that are depicted in Figs. 5 and 6 have not changed since 1990s. This may be explained by the fact that it becomes harder and harder to increase the a-index when it has already reached high values (similarly to what happens to the h-index in research citations). Another factor may be that not all Ph.D. graduates from the past 10–20 years have been added to the MathGenealogy dataset yet. However, as mentioned above, this “temporary” saturation behavior of collective a-indices is not the same as the one we observed for individual a-indices, since the production of Ph.D. graduates by a city or a country is not limited by the lengths of academic careers of individual advisors.
Further, we consider the evolution of $$a_1$$ and $$a_2$$ indices (along with their adjusted versions) of the top advisors listed in Tables 3 and 4. The respective plots are shown in Figs. 7, 8, 9, and 10. Note that the evolution of $$a_{\infty }$$-index is not depicted here, since the plots corresponding to all top advisors according to this index exhibit a highly similar pattern and thus would look indistinguishable in a figure.
Interestingly, from Figs. 7 and 8 one can observe that both the $$a_1$$ and the adjusted $$a_1$$ index values for all of the top advisors were very close to each other up to around 1970, which is a very recent date compared to the dates of their respective careers. However, in the past 3–4 decades, these indices have increased substantially and started to spread over a broader range of values, approximately between 6000 and 10,000. The ranking of advisors according to both the $$a_1$$ and the adjusted $$a_1$$ index has been stable over the past decades, with S.D. Poisson holding the top spot.
As for the evolution of $$a_2$$ and adjusted $$a_2$$ indices, there has been much more diversity and changes in the ranking of top advisors over the past decades (compared to $$a_1$$-index). As noted above, the $$a_2$$-index gives lower weights to distant descendants of an individual, which results in a lower order of magnitude of this index compared to the $$a_1$$-index. Nevertheless, despite a narrower range of values for this index, there have been several changes in the ranks of top advisors according to this index in the twentieth century (although many of these mathematicians worked in the nineteenth century). Notably, D. Hilbert has assumed the top spot in the $$a_2$$-index-based ranking only in the 1990s despite the fact that he received his own Ph.D. degree more than 100 years prior. Thus, the $$a_2$$-index can be viewed as a meaningful metric of an individual’s advising impact that lasts well beyond the end of one’s career and still can increase considerably in subsequent decades or even centuries.
As a concluding remark of this section, we should note that all of the aforementioned results should be viewed in the light of the fact that MathGenealogy dataset is not necessarily complete for the considered time interval, and some Ph.D. graduates (as well as some Ph.D. graduation year information, as mentioned above) may not have been added to the database yet. This may lead to discrepancies in the results presented here with those that may be obtained in future studies when more entries are added to the database. Nevertheless, the considered dataset is still rather large and comprehensive, and the presented results reveal interesting temporal patterns of the proposed individual and collective advising indices.

## Concluding remarks

We proposed a family of network-based advising impact metrics (a-indices) that are easy to calculate and interpret, as well as provided a flexible framework for quantifying advising impacts of individuals from different “eras” and stages of their academic careers, as well as collective advising impacts of countries and universities. Although we illustrated our approaches on MathGenealogy dataset only, these approaches are certainly applicable to other scientific domains where comprehensive advisor–student datasets may become available.
Due to the fact that we focus on the advising impact beyond the number of immediate students of an individual, this approach is not intended for measuring advising impacts of early-career scientists (simply calculating an out-degree for “young” advisors would still be a viable option). However, one may argue that a true impact of an academic advisor is evident towards later stages of career when one’s students achieve their own advising success. Therefore, we believe that these indices can be used in practical settings, for instance, by universities in order to quantify and promote individual and collective advising successes of their faculty members. This study shows the applicability of network-based techniques for these purposes. As one of the possible directions of future research, it could be of interest to look at “groups of influential advisors”, for instance, using optimization-based techniques that identify “central” groups of nodes in a network [12].
It should also be noted that this study is not intended to build direct comparisons or preferences between different metrics of advising impact, including those proposed here or those proposed in other related studies. Instead, we believe that long-term individual or collective advising impact should be considered in the context of an “ensemble” of various quantitative metrics, including the proposed a-indices. Similarly to debates regarding citation indices (e.g., whether the h-index or some other quantitative metrics of citations are the most appropriate to measure citation impact), there is no definitive answer to the question about the “best” metric for advising impact. We hope that this study will stimulate further research in this interesting research direction.

## Acknowledgements

This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute.

### Further information

Preliminary version of this paper appeared in: Chen X., Sen A., Li W., Thai M. (eds) Computational Data and Social Networks, Proceeding of CSoNet 2018, Lecture Notes in Computer Science, vol 11280, pp. 437–449, Springer, 2018.

### Competing interests

The authors declare that they have no competing interests.

## Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
print
DRUCKEN
Fußnoten
2
To be more consistent with the notation for the rest of the “a-indices” defined here, one may denote this index as $$a_0$$-index; however, for simplicity, throughout the paper we will call this metric the “a-index” (which may be viewed as an analogy to the h-index widely used as a citation metric).

3
Of course, the out-degree of a node, that is, the number of advised students, is the simplest measure that assesses immediate advising impact; however, it is not in the scope of this study as it does not reflect the advising impact of descendants.

Literatur
1.
Arslan E, Gunes MH, Yuksel M. Analysis of academic ties: a case study of mathematics genealogy. In: GLOBECOM Workshops (GC Wkshps), 2011. IEEE, 2011. p. 125–9.
2.
Malmgren RD, Ottino JM, Amaral LAN. The role of mentorship in protégé performance. Nature. 2010;465(7298):622. CrossRef
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Myers SA, Mucha PJ, Porter MA. Mathematical genealogy and department prestige. Chaos. 2011;21(4):041104. CrossRef
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Gargiulo F, Caen A, Lambiotte R, Carletti T. The classical origin of modern mathematics. EPJ Data Sci. 2016;5(1):26. CrossRef
5.
Taylor D, Myers SA, Clauset A, Porter MA, Mucha PJ. Eigenvector-based centrality measures for temporal networks. Multiscale Model Simul. 2017;15(1):537–74.
6.
Rossi L, Freire IL, Mena-Chalco JP. Genealogical index: a metric to analyze advisor–advisee relationships. J Inform. 2017;11(2):564–82. CrossRef
7.
Boldi P, Vigna S. Axioms for centrality. Internet Math. 2014;10(3–4):222–62.
8.
Marchiori M, Latora V. Harmony in the small-world. Physica A. 2000;285(3–4):539–46. CrossRef
9.
Jackson MO. Social and economic networks. Princeton: Princeton University Press; 2010. CrossRef
10.
Tsakas N. On decay centrality. BE J Theor Econ. 2016;19:1–18. MathSciNet
11.
Broido AD, Clauset A. Scale-free networks are rare. Nat Commun. 2019;10(1):1017. CrossRef
12.
Veremyev A, Prokopyev OA, Pasiliao EL. Finding groups with maximum betweenness centrality. Optim Methods Softw. 2017;32(2):369–99.
Titel
Network-based indices of individual and collective advising impacts in mathematics
verfasst von
Alexander Semenov
Alexander Veremyev
Alexander Nikolaev
Eduardo L. Pasiliao
Publikationsdatum
01.12.2020
Verlag
Springer International Publishing
Erschienen in
Computational Social Networks / Ausgabe 1/2020
Elektronische ISSN: 2197-4314
DOI
https://doi.org/10.1186/s40649-019-0075-0

Zur Ausgabe

format: plain


text:
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text:
# Conflict between polyglossia and ucharclasses

I have an issue with the following (Xe)LaTeX code where I want to get proper French spacing around the punctuation.

Here goes:

\documentclass{standalone}

\usepackage{polyglossia}
\setmainlanguage{french}
% \usepackage{ucharclasses}

\begin{document}
Ceci est un test: «avec des guillemets»
\end{document}

This yields a correct rendering:

However, if I uncomment the ucharclasses package, then the spacing gets incorrect:

I don't even need to load the Latin option of ucharclasses to get things wrong. I would love to not blame anyone and get things working, but have no idea where to start!

• You are loading the latin option as the default is to setup all groups. If you use e.g. \usepackage[Greek]{ucharclasses} everything is fine. Aug 12 '15 at 15:26
• Thank you Ulrike, I didn't know that point. As a matter of fact, in my complete document, I need to load the Latin block, otherwise the font doesn't switch back to my default font after I input characters from exotic blocks. That trick may be a workaround of a different issue I shall investigate though. However, why is that that loading Latin breaks the rules by polyglossia? Aug 12 '15 at 16:16
• Both uses the same xetex feature (classes). It is certainly possible to reset things, but one would need to investigate first. Aug 12 '15 at 16:20

The ucharclass package changes the class of the Latin characters, but polyglossia assumes this class is 0.

You can try with

\documentclass{standalone}

\usepackage{ucharclasses}
\usepackage{polyglossia}
\setmainlanguage{french}

\makeatletter
\appto\french@punctuation{%
\ifluatex\else
\XeTeXinterchartoks \BasicLatinClass \french@punctthin = {\nobreak\thinspace}%
\XeTeXinterchartoks \BasicLatinClass \french@punctthick = {\nobreakspace}%
\XeTeXinterchartoks \french@punctguillstart \BasicLatinClass = {\nobreakspace}% "«a" -> "« a"
\XeTeXinterchartoks \BasicLatinClass \french@punctguillend = {\nobreakspace}% "a»" -> "a »"
\XeTeXinterchartoks \LatinSupplementClass \french@punctthin = {\nobreak\thinspace}%
\XeTeXinterchartoks \LatinSupplementClass \french@punctthick = {\nobreakspace}%
\XeTeXinterchartoks \french@punctguillstart \LatinSupplementClass = {\nobreakspace}% "«a" -> "« a"
\XeTeXinterchartoks \LatinSupplementClass \french@punctguillend = {\nobreakspace}% "a»" -> "a »"
\fi
}
\makeatother

\begin{document}

Ceci est un test: «avec des guillemets»

\end{document}

• Looks great to me <3 Just a last question @egreg : as I checked gloss-french.ldf, it seems you replaced/appended according to the \z@ -> \BasicLatinClass pattern. But if I try to imagine the case 255 \french@punctthin, how can I be sure it works well and the space is well replaced by a \nobreak\thinspace. No need to specify \BasicLatinClass anywhere? It seems to work fine on a toy example, but I don't get why it would work as is with space and not with \z@... Aug 12 '15 at 20:19
• @xoolive I added also the Latin Supplement class. About class 255, there should be no problem, as class 0 is not involved. There are other problems, because the punctuation characters will not be recognized as Latin ones for transitions. Aug 12 '15 at 20:25
• As a matter of fact, after I am done with a Unicode block, I must pay attention not to use a comma (the fonts I use don't have punctuation) but use a Latin character first in order to force the switching back to Latin. So there's a way to fix this? ... Btw, I must have been confused, but what does class 255 stands for? Aug 12 '15 at 22:17
• @xoolive From the reference guide (texdoc xetex): “The boundary of a text string is considered class 255, wherever there is a boundary between a ‘run’ of characters and something else — glue, kern, math, box, etc.” Aug 12 '15 at 22:29


text:
# Equivalence Relations

Definition. Let $A$ be a nonempty set. Then $R\subseteq A\times A$ is called a binary relation on $A$. A binary relation $R$ on $A$ is said to be an equivalence relation if

1. $R$ is reflexive: $\forall x\in A$, $(x,x)\in R$
2. $R$ is symmetric: $(x,y)\in R\Longrightarrow (y,x)\in R$
3. $R$ is transitive: $(x,y)\in R\ \mathrm{and}\ (y,z)\in R\Longrightarrow (x,z)\in R$

Definition. Let $R$ be an equivalence relation on a set $A$. For each $x\in A$, let $$[x]=\{y\in A: (y,x)\in R\}$$ Then $[x]$ is called the ($R$-)equivalence class of $x\in A$.

Theorem 1. Let $R$ be an equivalence relation on a set $A$. Then

1. $\forall x,y\in A$, $[x]\cap [y]=\emptyset$ or $[x]=[y]$.
2. $A=\bigcup_{x\in A}[x]$.

Proof.

1. Let $x,y\in A$. If $[x]\cap [y]=\emptyset$, then we are done. Now suppose that $[x]\cap [y]\ne\emptyset$. Then $\exists z\in[x]\cap [y]$ so that \begin{align*}(z,x)\in R\ \mathrm{and}\ (z,y)\in R&\Longrightarrow (x,z)\in R\ \mathrm{and}\ (z,y)\in R\\&\Longrightarrow (x,y)\in R\end{align*} This implies that $[x]=[y]$.
2. Left as an exercise.

Definition. Let $A$ be a set. By a partition of $A$ we mean a family $\{A_i\}_{i\in I}$of nonempty subsets of $A$ such that

1. $\forall i,j\in I$, $A_i\cap A_j=\emptyset$ or $A_i=A_j$.
2. $A=\bigcup_{i\in I}A_i$

Theorem 1 says that given an equivalence relation $R$ on $A$, the $R$-equivalence classes form a partition of $A$. Conversely, given a partition $\{A_i\}_{i\in I}$, one can define an equivalence relation $R$ whose equivalence classes coincide with the $A_i$´s.

Theorem 2. Let $\{A_i\}_{i\in I}$ be a partition of a set $A$. Define a relation $R$ on $A$ as follows: $$\forall x,y\in A,\ (x,y)\in R\Longleftrightarrow x,y\in A_i\ \mbox{for some}\ i\in I$$ Then $R$ is an equivalence relation on $A$ and the $R$-equivalence classes coincide with the $A_i$´s.

Proof. Left as an exercise.

Given an equivalence relation $R$ on a set $A$, the set of all equivalence classes is called the quotient set of $A$ modulo (or mod in short) $R$ and is denoted by $A/R$ i.e. $$A/R=\{[x]: x\in A\}$$

There is a map $\gamma: A\longrightarrow A/R$ defined by $$\forall x\in A,\ \gamma(x)=[x]$$ $\gamma$ is called the canonical map from $A$ to $A/R$.

Example 1. (The Vector Space $\mathbb{R}^3$) Let $V$ be the set of all directed arrows in 3-space $\mathbb{R}^3$. Define a relation $\equiv$ on $V$ as follows: $$\forall \overrightarrow{AB},\overrightarrow{CD}\in V,\ \overrightarrow{AB}\equiv\overrightarrow{CD}\Longleftrightarrow\ \overrightarrow{AB}\ \mathrm{and}\ \overrightarrow{CD}\ \mbox{have the same direction and magnitude}$$ Then clearly $\equiv$ is an equivalence relation on $V$. Denote by $\mathcal{V}$ the quotient set $V/\equiv$ and each equivalence class $[\overrightarrow{AB}]\in\mathcal{V}$ is called a vector. Each directed arrow $\overrightarrow{AB}$ whose starting point is $A$ and terminal point is $B$ is $\equiv$-related to a directed arrow $\vec{a}$ whose starting point is the origin $O=(0,0,0)$ and terminal point is $B-A$. $\vec{a}$ can be identified with its terminal point $B-A$ which is an ordered triple $(a_1,a_2,a_3)$. Hence from here on, without loss of generality, we may assume that each equivalence class is represented by such a vector. Define the vector addition $+:\mathcal{V}\times\mathcal{V}\longrightarrow\mathcal{V}$ by $$\forall [\vec{a}],[\vec{b}]\in\mathcal{V},\ [\vec{a}]+[\vec{b}]:=[\vec{a}+\vec{b}]$$ Here, $$\vec{a}+\vec{b}=(a_1,a_2,a_3)+(b_1,b_2,b_3)=(a_1+b_1,a_2+b_2,a_3+b_3)$$ Also define the scalar multiplication $\cdot :\mathbb{R}\times\mathcal{V}\longrightarrow\mathcal{V}$ by $$\forall c\in\mathbb{R},\ \forall [\vec{a}]\in\mathbb{V},\ c[\vec{a}]:=[c\vec{a}]$$ Here, $$c\vec{a}=(ca_1,ca_2,ca_3)$$ Let $[\vec{a}]=[\vec{a}’]$ and $[\vec{b}]=[\vec{b}’]$. Then $\vec{a}=\vec{a}’$ and $\vec{b}=\vec{b}’$ i.e. $a_i=a’_i$ and $b_i=b’_i$, $i=1,2,3$. Thus $$\vec{a}+\vec{b}=\vec{a}’+\vec{b}’\Longrightarrow [\vec{a}]+[\vec{b}]=[\vec{a}+\vec{b}]=[\vec{a}’+\vec{b}’]=[\vec{a}’]+[\vec{b}’]$$ Hence, the vector addition is well-defined. It can be shown similarly that the scalar multiplication is also well-defined. We will see in the next example that there is a bijective from $\mathcal{V}$ to $\mathbb{R}^3$.

Example 2. (The Kernel of a Function) Let $f: A\longrightarrow B$ be a function. The kernel of $f$, $\ker f$ is defined by $$\ker f=\{(x,y)\in A\times A: f(x)=f(y)\}$$ It is easy to see that $\ker f$ is an equivalence relation on $A$. It turns out that the quotient set $A/\ker f$ coincides with the partition of $A$ by the pre-images $f^{-1}(z)$, $\forall z\in A$. (See Problem Set 12 #5 (b).) Now we suppose that $f: A\longrightarrow B$ is surjective (onto). Define a map $\psi: A/\ker f\longrightarrow B$ by $$\forall [x]\in A/R,\ \psi([x])=f(x)$$ Then $\psi$ is a bijective and $\gamma=\psi\circ f$, where $\gamma: A\longrightarrow A/\ker f$ is the canonical map. (Its proof is left as an exercise.)

In Example 1, we have seen that given a directed arrow $\overrightarrow{AB}$ there is a $\equiv$-related directed arrow whose starting point the origin $O$ and terminal point $B-A$ and that such a directed arrow can be identified with its terminal point $\vec{a}:=B-A=(a_1,a_2,a_3)\in\mathbb{R}^3$. This defines a surjective map $f: V\longrightarrow\mathbb{R}^3$ and $\ker f=\equiv$. Therefore, we see that there is a bijective $\psi$ from $V/\ker f=V/\equiv$ to $\mathbb{R}^3$ defined by $$\forall [\vec{a}]\in V/\equiv,\ \psi([\vec{a}])=(a_1,a_2,a_3)$$ Furthermore, this bijective is a vector space homomorphism i.e. it preserves the vector addition and the scalar multiplication: \begin{align*}\psi([\vec{a}]+[\vec{b}])&=\psi([\vec{a}])+\psi([\vec{b}])\\\psi(c[\vec{a}])&=c\psi([\vec{a}])\end{align*} Therefore $V/\equiv$ is isomorphic to $\mathbb{R}^3$ as a vector space.

Example 3. (The Construction of Rational Numbers From Integers) Let $\mathbb{Z}$ be the set of all integers. Define a relation $\sim$ on $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})$ by $$\forall (a,b),(c,d)\in \mathbb{Z}\times(\mathbb{Z}\setminus\{0\}),\ (a,b)\sim (c,d)\Longleftrightarrow ad=bc$$ Then $\sim$ is an equivalence relation. (Its proof is left as an exercise.) The equivalence class of $(a,b)$ is denoted by $\frac{a}{b}$. For example, $(1,2)\sim (2,4)$ thus $\frac{1}{2}=\frac{2}{4}$. We see that the quotient set $\mathbb{Z}\times(\mathbb{Z}\setminus\{0\})/\sim$ can be identified with $\mathbb{Q}$, the set of all rational numbers.

Example 4. (Residue Classes Modulo $n$) Let $\mathbb{Z}$ be the set of all integers. Define a relation $\equiv\ \mathrm{mod}\ n$ on $\mathbb{Z}$ by $$\forall p,q\in\mathbb{Z},\ p\equiv q\ \mathrm{mod}\ n\Longleftrightarrow n|(p-q)$$ Then $\equiv\ \mathrm{mod}\ n$ is an equivalence relation. (Its proof is left as an exercise.) An integer $m$ can be written as $m=nk+r$ where $k$ is the quotient and $0\leq r<n$ is the remainder when $m$ is divided by $n$. Thus $m\equiv\ \mathrm{mod}\ n r$. This means that there are exactly $n-1$ equivalence classes $$[0],[1],\cdots,[n-1]$$ These equivalence classes are called the residue classes modulo $n$. The quotient set $\mathbb{Z}/\equiv\ \mathrm{mod}\ n$ is usually denoted by $\mathbb{Z}_n$. We can define addition and multiplication on $\mathbb{Z}_n$ as follows: $\forall [a],[b]\in\mathbb{Z}_n$, \begin{align*}[a]+[b]&=[a+b]\\ [a]\cdot [b]&=[ab]\end{align*} These operations are well-defined because the equivalence relation $\equiv\ \mathrm{mod}\ n$ preserves the operations, namely if $a\equiv b\ \mathrm{mod}\ n$ and $c\equiv d \mathrm{mod}\ n$ then \begin{align*}a+c&\equiv b+d\ \mathrm{mod}\ n\\ ac&\equiv bd\ \mathrm{mod}\ n\end{align*} An equivalence which preserves operations are called a congruence relation. Congruence relations play a crucial role in constructing quotient algebras. For more details about congruence relations, see for example the reference [3]. Note that the equivalence relations in Examples 1 and 3 are also congruence relations. $(\mathbb{Z}_n,+)$ is an abelian group and $(\mathbb{Z}_n,+,\cdot)$ is a commutative ring with unity. $(\mathbb{Z},+,\cdot)$ is a field if $n$ is a prime.

Example. (Digraphs) Let $G=(V,A)$ be a digraph. Let $\sim$ be mutual reachability relation on $V$, i.e. $\forall x,y\in V$, $x\sim y$ if and only if $x$ and $y$ are mutually reachable. It can be easily seen that mutual reachability relation is an equivalence relation. The equivalence classes are called the strongly connected components or simply strong components of the digraph $G$ and they are subgraphs of $G$.

Theorem. Let $G=(V,A)$ be a digraph and let $T$ be the partition of $V$ into the strong components of $G$. Construct a new digraph $G’=(T,A’)$, where $\forall X,Y\in T$, $X\to Y\in A’$ if and only if $X\ne Y$ and $x\to y\in A$ for some $x\in X$ and $y\in Y$. Then $G’$ is a DAG.

Proof. Suppose that there is a cycle in $G’$, $x_0\to x_1\to\cdots x_k\to x_0$ for some $x_0,\cdots,x_k$ belonging to different strong components of $G$. But all theses vertices are mutually reachable in $G$ so this is a contradiction. Therefore $G’$ is acyclic.

References.

[1] Essential Discrete Mathematics for Computer Science, Harry Lewis and Rachel Zax, Princeton University Press, 2019

[2] Set Theory, Charles C. Pinter, Addison Wesley Publishing Company, 1971

[3] A Course in Universal Algebra, Stanley N. Burris and H. P. Sankappanavar, Graduate Texts in Mathematics, Springer-Verlag, 1981. The book is available online for free at the first named author’s web page here.


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## FANDOM

707 Pages

 “ Fight my magic all you want; I like the struggle. No matter, I'll win and you'll stay by my side forever! ” –Widow flaunting her power

Widow is the fifth demon boss in Shadow Fight 2, first appearing in Act V: The Greatest Temptation.

## Appearance

Widow closely resembles a Maiko Geisha, with a few exceptions. She has a Shimada hairstyle held together with hairpins, and a red Kimono which has sleeves underneath that resemble bandages. She does not have the white makeup on her face like the traditional Maiko, only having a red dot on her forehead, which is similar to the Bindi of Indian culture.

## Storyline

### Sensei's Story Part 5. Humiliated and Insulted

After Wasp is defeated, she tells Sensei what actually happened; Widow is a woman who fell in love with the Prince. However, her love is unrequited and Prince rejected her. This broke Widow's heart and caused her to want to rule all men and play them off, making them fight each other as a way of her revenge. Widow starts by manipulating Shogun, the Prince's warlord, to make him overthrow Prince. Widow promised Shogun a magical Sphere that will turn him into a great warlord, but will also cause him to become her loyal slave.

When Sensei arrives at Widow's place, Widow becomes angry seeing him and wants to make him suffer, as she does not believe in men anymore. Widow's followers proceed to attempt to do the job, but they are defeated by Sensei. Before Widow proceeds to fight Sensei, she states that she will strike back for what the Prince has done to her and that Sensei will deliver this message to him. Sensei, however, defeats her and manages to speak some sense into her, saying that he sympathizes with her. Realizing that Sensei is honest with his intention to help, Widow informs him that Prince had come and seized the Sphere from her. Shogun's attempt to stop it was useless and Prince took him to be executed in the place where he had first met Widow. She also warns Sensei that the Prince has become wild after he got hold of the Sphere.

### Act V: The Greatest Temptation

After the defeat of Wasp, Shadow and the group continue their journey to get the remaining seals. Sensei tells Shadow that this is not their war, but Widow seems to be a worthy opponent and claims that they should pay her a visit. The group then moves on and they eventually meet Widow. Upon eyeing her, Sly feels a strong attraction towards her. Sensei, however, is strong enough to resist it. Unusually for Widow however, upon seeing Shadow, who is unaffected by her, she is impressed by him and claims she will keep an eye on him.

The story then goes on where Shadow defeats her bodyguards and eventually battles her and wins, and she hands him the Orange Seal.

Widow and the rest of the demons team up to stop Shadow from closing the Gates of Shadows. She battles Shadow fifth in line.

### Interlude

Shadow, on his quest to break the seals, encounters Widow again in Act V of Interlude where she attempts to stop Shadow from reopening the Gates of Shadows. Before the fight, Widow tells Shadow that Titan's minions are different from the enemies Shadow has faced so far and that he would not be able to defeat them. After defeat, Widow tells Shadow that Titan has been leading him from the very beginning like a puppet master and tells him that even now he is doing what Titan wishes.

## Personality

Widow is a powerful warrior whose beauty mesmerizes all men who get a glimpse of her. Sly, when he sees her for the first time, becomes infatuated with her and tries getting her gifts like gems and stones. Even Sensei feels the pull of her magic, but his years of martial arts mastery and discipline keep him from this temptation. Widow's first four bodyguards were in fact suitors enthralled by her charm. They fight Shadow because they thought he was trying to get Widow's attention. The only man unaffected by her beauty was Shadow himself. Widow is never easily pleased, however Shadow does catch her eye. She is not as aggressive as most of the other demons, barring Hermit.

## Bodyguards

Widow's bodyguards are mainly men who have been charmed. The only exception is Puma, a woman who serves Widow to help her achieve her ambitions. Most bodyguards attack Shadow in an attempt to eliminate the competition, because they are mistaken in the belief that Shadow desires Widow as they do.

1. Irbis: bodyguard armed with Steel Batons
2. Wolf: bodyguard armed with Swords
3. Capra: bodyguard armed with Silver Spear
4. Bear: bodyguard armed with Heavy Hammer
5. Puma: bodyguard armed with a Sentinel Kusarigama

Bear and Puma's weapons are unobtainable, though the sillouette of Bear's hammer is exactly the same as the Northern Hammer. Strangely though, it's Super Slash is the same as the Dadao.

## Equipment

Widow uses a pair of fans as weapons. They work similar to sword-type weapons, but have a shorter range and a different super-slash.

For ranged weapons, Widow first uses Ghostly Kunai, then switches to Assassin's Daggers after facing defeat in one round.

For magic, she uses Energy Ball, Ice Ball (after losing once) and Ice Pillar (after losing twice).

When fought in Gates of Shadows and Interlude, she uses Assassin's Dagger and Ice Pillar.

## Perks and Enchantments

• Teleportation

Widow has the ability to teleport; she disappears in the cyan beams coming from her fans and shows up attacking from the back. It can be dodged using any movement combo. Players also can interrupt the attack by hitting Widow before she finishes her teleport. Widow uses this perk after losing once.

• Lifesteal

Widow's Fans and ranged weapons are enchanted with Lifesteal, giving Widow a chance on every successful hit to replenish her health by 250% of the damage dealt to the player.

• Regeneration

Widow's armor and helm are enchanted with Regeneration, giving a chance to regenerate 4.5% of her health for 5 seconds after being hit. Appears after Widow loses twice.

## Quotes

• Well, what do we have here? An interesting specimen... I've met so many men, so boring, so average, but... There's something peculiar about this one. I'll keep my eye on you. - Widow commenting about Shadow
• Such a powerful young man – you'll be a perfect addition to my collection.
• I'll be here when you're ready to try again. A worthy adversary such as yourself... we could be good friends. Or perhaps more? - Widow won
• How does it feel to defeat a woman? And you're the one who'll close the Gates of Shadows? You have no idea how quickly their power will tear you apart! - Widow defeated
• Don't think you've seen the last of us. We'll meet again soon, Shadow. Until then, enjoy your victory. - Gates of Shadows, after defeated

### Interlude

• Titan's minions are like nothing you've ever seen before. You think you're prepared? It's never enough! - Interlude, before fight
• Titan's been leading you from the very beginning! Pulling your strings like a puppet master. Even now you do what he wishes. - Interlude, after defeated

### Old Wounds

• Are you coming to humiliate me the way Prince did? You males are all the same...
• He's been telling me of your kindness, honor and dignity. But I don't believe any male's words anymore. Each of you shall suffer!
• Prince humiliated me by rejecting and laughing at my feelings. Now I shall strike back! You are his friend, so you'll deliver this message to him...
• There is no way back for me. I see that you want to help, but it's too late-------Prince took the sphere already. Shogun was trying to defend it, but in vain.
• I've never seen Prince so wild. His eyes were flaming! He thought it to be funny to execute a traitor in the place where we first met. I'll show you...

## Reward

• Normal:
• 277 green orbs
• 80 red orbs
• Eclipse:
• 3652 green orbs
• 1689 red orbs
• 48 purple orbs

### Eclipse Mode reward

Players can obtain Widow's Fans by defeating her once more in Eclipse mode. It can be obtained at any level, as early as level 31. The fans are enchanted with Lifesteal. The formula for the power of the Lifesteal enchantment is as follows:

$\frac{3639}{100} \times \mathrm{level} + 105$

### Achievements

• No Heart to Break

Awarded for defeating Widow.

• Frozen Heart

Awarded for winning a fight with Widow once more in Eclipse mode.

## Soundtrack

During the battle with Widow and her bodyguards, the track “Shadow Lady” plays.

## Trivia

• Widow's Ice Ball is one of her magic attacks that is not obtainable. As of Christmas 2016 update, players can however obtain the Shard of Eternal Ice magic which functions in a similar manner.
• Widow's fans (and other fans weapons) are one of the only weapons that can be thrown in fights. The others are Staff weapons.
• Her bodyguards do not wield any weapons that are from Act V.
• Widow is the only demon who has somewhat affectionate feelings for Shadow.
• It is later revealed that Shadow looked very similar to the Prince, whom Widow once fell in love with.
• Her counterpart in Shadow Fight 1 is Huntress.

## Start a Discussion Discussions about Widow

• #### How to defeat widow in SF2

13 messages
• What about battle hammers I have ☠
• Use butchers knives and dark implosion magic and armor n helm with regeneration or dmg absorbsion
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