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Značka: Taxes
Crypto taxation could deter investors, says Thai ruling party MP
podľa Cointelegraph By Prashant Jha | jan 21, 2022 | 0 |
Thailand's Committee on Monetary Affairs, Finance, Financial Institutions and Financial Market conducted a virtual meeting to discuss various aspects of crypto taxation.The ruling party MP Watanya Wongopasi posted a summary of the discussion on her Facebook page and urged the Excise department to do their due diligence before imposing any tax on the crypto trading market.In the meeting, Paiboon Nalinthrangkurn, the chairman of the Federation of Thai Capital Market Organisations noted that a tax on stock trading and digital asset trading could decrease market liquidity by 40%. He also warned that heavy taxation would deter foreign and small investors from trading.Yutthana Srisavat, the CEO and founder of iTax proposed a corporate tax or a value-added tax instead of imposing a trading tax. He also made it clear that the decentralized nature of crypto makes it extremely difficult to gather buyer and seller information, making it near impossible to collect tax information.The Thai Excise department noted that the majority of its focus has been on taxing the stock market, and it has made little progress in terms of crypto trading taxation. However, the department assured that they are carefully studying the crypto market and the taxes will be only imposed after careful consideration.Chonladet Khemarattana, president of the Thai Fintech Association advocated for a free market to compete with other nations. He urged the government to monitor the growth of the crypto ecosystem in the short term before moving on with tax implementation.Related: Former Thai SEC chief lays out three critical issues with crypto taxationsCrypto taxation has become a hot topic in Thailand especially after the government proposed a 15% tax on crypto gains. Several current and former executives have come out to warn against the proposal, including former Thai SEC executive Tipsuda Thavaramara. As Cointelegraph reported earlier, Thai Prime Minister Prayut Chan-o-cha has instructed the revenue department to offer clarification for investors and the public on crypto taxation soon.
Crypto-focused software firm Lukka raises $110M, reaches $1.3B valuation
podľa Cointelegraph By Turner Wright | jan 14, 2022 | 0 |
Cryptocurrency accounting and data company Lukka said it had raised $110 million in a new funding round led by hedge fund giant Marshall Wace.In a Friday announcement, Lukka said it had completed a $110 million Series E funding round led by Marshall Wace, with participation from Soros Fund Management — a fund created by billionaire investor George Soros — Liberty City Ventures, S&P Global, and accounting advisor CPA.com. Lukka said it planned to use the funds for "aggressive growth and global expansion strategy" with its current customer base dealing in derivatives, decentralized finance, and other products related to the crypto space.Lukka CEO Robert Materazzi said the crypto ecosystem had entered a "new phase of maturity," with the firm aiming to address challenges from innovative technology in the industry. The company added the S&P Dow Jones Indices, major accounting firm RSM, and financial services company State Street to its list of customers in 2021.The crypto-focused software firm announced a $53 million raise in March 2021, and recently achieved unicorn status with a more than $1 billion valuation. Serving many active crypto funds, Lukka has reportedly processed $2.1 trillion in unique transactions to date.Related: Lukka Co-CEO Explains How Blockchain Data Saves on TaxesFounded in 2014, the New York-based company focuses on offering blockchain and tokenized asset data to help businesses involved in the crypto space. Among Lukka's products are tax solutions and data valuations to a digital asset content library.
Coinbase, PayPal join TaxBit Network for free crypto tax forms
podľa Cointelegraph By Helen Partz | jan 11, 2022 | 0 |
Crypto tax compliance firm TaxBit is working to unite major industry companies like Coinbase and Binance.US within one network to enable free 2021 tax forms for users.The crypto tax software provider on Tuesday launched the TaxBit Network, a certified network aiming to democratize crypto tax calculations and tax forms by providing necessary tax reporting data to all network users for free.The TaxBit Network at launch includes about 20 major crypto-related businesses in the United States, including firms like PayPal, Binance.US, Coinbase, FTX.US, Gemini, Celsius Network, Blockchain.com, Venmo, Paxos, OkCoin and BlockFi. The network will be adding new companies daily, TaxBit's VP of marketing Michelle O'Connor told Cointelegraph.As part of the initiative, every user of a certified TaxBit network company will be able to receive free and accurate 2021 tax forms. Additionally, a number of participating platforms will be incorporating a quick TaxBit sign-up within their apps to simplify access to tax reporting tools."If a user has taxable transactions on platforms not part of the TaxBit Network, a cost will apply to retrieve tax forms from out-of-network platforms," TaxBit noted.The new development aims to eliminate barriers to mainstream crypto adoption by ensuring that crypto users in the U.S. are able to stay tax compliant without spending tons of money. "Historically, the process of generating cryptocurrency tax forms cost individuals anywhere from hundreds to thousands of dollars per year depending on whether they used do-it-yourself software or enlisted an accountant," the announcement notes."Our portfolio performance and tax optimization solutions empower year around decisioning versus that tax season-only mentality. With market volatility so prevalent, we provide the retail consumer with tax compliant solutions to optimize their holdings through tax-loss harvesting," O'Connor stated.Related: US lawmaker planning to introduce comprehensive crypto bill in 2022Binance.US CEO Brian Shroder pointed out that the company's participation in the TaxBit Network is a crucial step in their commitment to safety and compliance, particularly after President Joe Biden signed the infrastructure bill into law in November.TaxBit is a major crypto-focused tax compliance firm that brings together tax attorneys and software developers to build software to simplify and automate the process of crypto tax reporting. Backed by Winklevoss twins' family office, TaxBit has seen notable growth recently, securing a $130 million raise at a $1.3 billion valuation in August 2021.
Crypto trades in Thailand now reportedly subject to 15% capital gains tax
podľa Cointelegraph By Helen Partz | jan 6, 2022 | 0 |
The government of Thailand is progressing in regulating the local cryptocurrency ecosystem by reportedly enacting new tax rules for the industry.Profits from crypto trading in Thailand are now subject to a 15% capital gains tax, The Bangkok Post news agency reported Thursday.The Thai Revenue Department also plans to step up its monitoring duties following a booming digital asset market last year. The department has the authority to collect taxes from crypto trades as profits from such activity are considered assessable income under Section 40 of the Royal Decree amending Revenue Code No.19, the report stated.The finance ministry recommended investors to calculate and report their income from cryptocurrencies in tax declarations in 2022 to avoid legal penalties. The new tax will be collected from all taxpayers who gained profits from crypto, including trading and mining operations.On the other hand, cryptocurrency exchanges are reportedly exempt from new tax requirements.Akalarp Yimwilai, co-founder and CEO at major local exchange Zipmex Thailand, raised concerns about the ongoing uncertainty regarding the crypto tax reporting process and how to calculate profits."Tax methods and calculations should be more concise, clear and easy to understand. Many people I know want to pay taxes, but don't know how to calculate them," Akalarp said.Related: Crypto mining reportedly rises in Thailand due to Chinese crypto banThe new report comes in line with the Thai government's plans to define "red lines" for crypto in early 2022. Bank of Thailand governor Sethaput Suthiwartnarueput officially announced in mid-December that the central bank was planning to release new regulations specific to the crypto industry early this year.As previously reported by Cointelegraph, financial authorities in Thailand have been considering legislation to collect a 15% capital gains tax on crypto since at least March 2018.
Indian taxman recovers $6.62M from WazirX for evading tax on commission
podľa Cointelegraph By Arijit Sarkar | jan 1, 2022 | 0 |
Indian crypto exchange WazirX has reportedly paid over $6.6 million (49.2 crore rupees) following non-payment of Goods and Services Tax (GST) on trade commissions. The total recovery includes the pending tax of $5.43 million (40.5 crore rupees), the interest and a penalty for non-payment.Government officials from the Central GST and Central Excise committee (CGST Mumbai Zone) recovered the funds from the crypto exchange after detecting a GST evasion of $5.43 million on the commissions. A typical GST fraud involves creating fake invoices without actually moving the goods between the seller and the buyer.Officers of CGST Mumbai East comm'te have detected GST Evasion of Rs 40.5 Cr. on commission of Wazir X Crypto Currency & recovered Rs 49.2 Cr. in cash as GST, interest & Penalty today on 30.12.2021 from Zanmai Labs Pvt. Ltd. @nsitharamanoffc @mppchaudhary @cbic_india @PIBMumbai— CGST Mumbai Zone (@cgstmumbaizone) December 30, 2021According to local media Economic Times, the tax department detected that WazirX uses its in-house WRX tokens for commissions, which were distributed by Zanmai Labs. Further investigation revealed that the crypto exchange missed out on paying 18% tax on the total tokens issued based on its market price.The investigators revealed that WazirX paid GST on the 0.2% commission it charges users for making trades with local currency i.e. the rupee, clarifying:"But in cases where the trader opts for transaction in WRX coins, the commission charged is 0.1% of trading volume and they were not paying GST on this commission."It is also important to note that WazirX and WRX tokens are owned by Binance, the world's biggest crypto exchange in terms of the trading volume. According to a Zanmai Labs spokesperson, the non-payment of tax was related to the misinterpretation of GST rules:"We voluntarily paid additional GST in order to be cooperative and compliant. There was and is no intention to evade tax."WazirX CEO Nischal Shetty previously told Cointelegraph about the importance of regulatory clarity for retail adoption. He also warned that an overnight regulation may harm the progress of the crypto ecosystem and leave open loopholes for bad actors:"There is a $2.5-trillion market out there, and it is not going to wait for any nation to come on board. I've been tweeting '#IndiaWantsCrypto' for over 1,000 days with the sole objective of having crypto regulation in India."Day 1000What a milestone for Indian Crypto!With #IndiaWantsCrypto my mission has been:- Bring positive crypto regulation in India- Spread right information about CryptoLakhs of people have joined this campaignLet's continue our missionJai Hind #IndiaWantsCrypto— Nischal (WazirX) ⚡ (@NischalShetty) July 28, 2021
While the concept of GST is fairly new in the region, the government of India has previously agreed to show leniency to defaulters and fraudsters — typically settling such cases with a monetary penalty and a lower probability of jail time. WazirX has not yet responded to Cointelegraph's request for comment.Related: Indian trade group recommends 'special class security' status for cryptoIn an attempt to help the Indian government decide crypto laws, the Confederation of Indian Industries (CII) proposed to treat cryptocurrencies as securities of a special class. A report released by the non-government trade association showed the CII proposes to formulate new regulations around the nascent crypto market instead of regulating them under existing securities law.As Cointelegraph reported, the CII recommended a special provision of income tax and GST laws, which will treat cryptocurrencies as an asset class for tax purposes unless specifically treated as "stock in trade" by a participant.
Spreading holiday joy through charitable giving with cryptocurrency
podľa Cointelegraph By Rachel Wolfson | dec 25, 2021 | 0 |
The holidays are the perfect time of the year for giving back, and the rise of cryptocurrencies has created even more opportunities for charitable initiatives. This was highlighted during Giving Tuesday 2021, the Tuesday after Thanksgiving which saw over $2.4 million raised in cryptocurrency from the nonprofit fundraising platform The Giving Block.As crypto philanthropy becomes a new subsector of the cryptocurrency economy, some in the industry believe that crypto donations will only continue to increase. Alex Wilson, co-founder of The Giving Block, told Cointelegraph that last year the organization raised about $4 million in crypto donations, noting that this year, over $100 million in crypto donations will likely be received.According to Wilson, this growth is partly due to the fact that donating crypto is more tax efficient than fiat donations. "Anyone who donates before December 31 is able to claim a deduction for the 2021 tax year. This is a great way to offset some of your gains." Wilson added that over 1,000 nonprofits currently accept crypto donations through The Giving Block, a few of which include St. Jude, Save the Children and United Way. "Next year, we have a lot of partnerships that are going live and we expect our growth to accelerate. We're estimating that we'll process nearly $1 billion in donations next year and work with over 6,000 nonprofits."Cryptocurrency giving campaigns for the holidaysWhile a number of nonprofits have started to accept crypto donations, it's also notable that campaigns centered around cryptocurrency philanthropy are being launched this holiday season. For example, in early December, Upbring Innovation Labs — a Texas-based organization seeking to advance technology in the nonprofit sector — launched the Give Big TX Crypto Fund. Ryan Park, vice president of innovation of Upbring, told Cointelegraph that the fund is a joint cryptocurrency campaign working with twelve Texas-based nonprofits: "You can think of this as a 'cause fund.' The cause here is to make Texas a better place to live. This is also about showing nonprofits that they can adopt new Web 3.0 technologies to advance. The larger goal overall is to see Texas emerge as a leader in crypto philanthropy." Park shared that the Give Big TX Crypto campaign is partnering with organizations including Austin Pets Alive, Big Brothers Big Sisters Lonestar, Catholic Charities of Central Texas and eight other nonprofits. He added that the Texas Blockchain Council — a 501 C (6) organization — is also part of this initiative given the group's involvement in advancing blockchain throughout different industries in Texas. Kelsey Driscoll, senior innovation program strategist at Upbring, further told Cointelegraph that the campaign will be accepting over 40 different types of cryptocurrencies for donations through Dec. 31, all of which are facilitated by The Giving Block. "When donations are made, The Giving Block automatically converts them to United States dollars, so accepting crypto has been just as easy as accepting fiat donations, if not easier," she remarked. Driscoll added that the subreddit group r/Bitcoin will be matching Bitcoin (BTC) donations when contributions are made to any of the charities supported by The Giving Block. Pawthereum, a decentralized community-run project supporting animal shelters, has also launched a charitable cryptocurrency campaign this month. John Weathers, community manager for Pawthereum, told Cointelegraph that its 12 Days of Crypto Giving campaign allows for crypto donations to be made for specific projects that help animals in need. The Pawthereum project was created as a fork of the meme cryptocurrency project Grumpy Cat Coin, which raised $70,000 in crypto funds for the Sterling Animal Shelter in Massachusetts. Most recently, Pawthereum raised $25,000 through crypto donations for Muttville Senior Dog Rescue, a San Francisco-based animal shelter caring for dogs with special needs. According to Weathers, close to $400,000 worth of crypto has been donated since the campaign was launched on Dec. 14.Related: Is crypto a boys' club? The future of finance is not genderedNonfungible tokens, or NFTs, are also being leveraged for donations this year. Given that the market growth for NFTs sales is expected to reach $17.7 billion by the end of 2021, this sector is launching one of the largest crypto charity events this season. Known as Right-Click, Give!, this is an auction open to the public hosted on the NFT platform Opensea. The auction ran through Dec. 24, and all proceeds will be donated to Blankets of Hope, a charity that provides warm blankets to the homeless while also teaching kindness to children in school. Mike Fiorito, co-founder of Blankets of Hope, told Cointelegraph that as an avid NFT collector, he is well aware of how welcoming the NFT community is as a whole. As such, he believes that more NFT-focused charitable campaigns will emerge. "There are a lot of kind people in the NFT space that are making fortunes — no matter how big or small — and want to give back," he said. Park also pointed out that the Give Big TX Crypto campaign is allowing NFT artists to work directly with nonprofits to donate proceeds earned from minting nonfungible tokens. "Many artists doing NFT drops are looking to work with nonprofits and this is an opportunity to do so. We have two NFT projects donating proceeds from their mints to our fund." Will charitable crypto campaigns catch on?Although there are currently only a handful of charitable crypto campaigns present, the benefits associated with cryptocurrency donations may result in mainstream adoption moving forward. While U.S. donors don't have to pay capital gains taxes on any crypto assets they donate to a registered nonprofit, there are other technological advantages. For instance, Nawzad Amiri, community leader for Pawthereum, told Cointelegraph that the transparency provided by a blockchain network, along with the speed of transactions, is impressive when it comes to crypto donations versus fiat. Moreover, statistics from The Giving Block found that crypto donors may be willing to contribute more to charity, noting that $11,000 is the average cryptocurrency donation size on The Giving Block. Donation data from Giving USA found that $737 was the average charity donation for Americans in 2020. While the benefits may be clear to some, education seems to be the biggest challenge hampering adoption. For example, although Texas is growing into one of America's biggest crypto capitals due to its friendly stance toward blockchain and mining power capabilities, Park shared that it has been challenging to bring Texas nonprofits on to the Give Big TX Crypto campaign: "We reached out to about 60 nonprofits and are partnering with 12 total. It seems like this would have been a shoo-in but there is still a long way to go in terms of educating the world on the potential of crypto philanthropy."Another challenge worth mentioning is that while NFT's may appear as ideal donation assets, there is uncertainty regarding tax deductions. Fiorito explained that he is still trying to determine if donating NFTs is considered a tax-deductible event. "This is a cloudy area because we are so early in the NFT space," he commented. Due to this uncertainty, the Right-Click, Give! Auction is also accepting cryptocurrency donations through The Giving Block. Related: Biggest NFT drops and sales in 2021Challenges aside, it's clear that crypto philanthropy has opened a new door of opportunities to a generation eager to give back. For instance, Park pointed out that the donor base for many of the nonprofits partnering with the Give Big TX Crypto campaign is over the age of 70. According to statistics from The Giving Block, the average age of crypto users is 38 years old, as Wilson added:"The people donating here are individuals that live and breathe Web 3.0 or that have held crypto for a long time. A big piece of this now is just educating those people that this opportunity exists."


The team is multidisciplinary, gathering competences in environmental/occupational epidemiology, environmental toxicology, biology and exposure/analytical sciences.
This axis relies on the PARIS birth cohort follow-up and investigates the natural course of allergic/respiratory diseases and their association with lifestyle and indoor/outdoor environment in childhood.
Echoing PARIS cohort research, this axis aims to study health effects of transport professionals' exposure to air pollutants. In addition, it produces knowledge on occupational allergy and assess the effectiveness of public policies, through national and international networks.
To study the biological plausibility of epidemiological findings, the team develops in vitro models to assess cellular and molecular effects related to air pollutants.
Particulate matter being a great public health concern, particularly in respiratory health, cellular and molecular impact of exposure to Particulate Matter PM2.5 from various sources, industrial or traffic is assessed on reconstituted nasal epithelium, in an adapted in vitro model.
Particulate matter being a great public health concern, particularly in respiratory health, cellular and molecular impact of exposure to particulate matter originating from fumigants (ANR ASTRID – FUMITOX project), is assessed on reconstituted nasal epithelium, in an adapted in vitro model.


Hope the weather if perfect for the next event!
Many thanks! So far, the forecast is good.
Glad the day wasn't a total washout! Good luck next time.
Wow – That is one snazzy setup, you guys. It looks great! Keep on working it!
A steam punk festival sounds like fun.
What is a steam punk, and how are they festive?
Here is Merriam-Webster's definition: "science fiction dealing with 19th-century societies dominated by historical or imagined steam-powered technology." Lots of fun!
you always find the positive even on a rainy day. Glad you found some side benefits, and good luck next weekend.
Unless the occasion is really dire—and sometimes it is—there is always something good to be gleaned.
A Steam Punk Fair sounds fab – I wish I could come! I do hope you have a successful time and Maya is sent off to far flung places once again.
Love that piece of Art you bought. It would look good as fabric.
I just found Erik Howell on Facebook, was hoping he had a website but I couldn't see a link.
I know. What's up with that? I looked for a website to link into my piece, but I couldn't find one either.
Yes, the rain definitely kept people away. But we had a good time anyway.
What a shame about the rain… never mind the Steam Punk festival sounds promising…. is your own Snow Gauge going as punk?
Oh, dear, oh, dear–what a drag! We were having a yard sale and had to move everything inside real quick, too, but our junk isn't as susceptible to rain as your books would be!
Ah, well! We still had a good time.
Nice setup! I like the canopy safeguard.
Thanks, Alice. A requirement at a lot of outdoor events.
Lots of success for the next event! Your canopy surely looks good, so sparkling white and a perfect rain catcher.
About books getting wet, I like to read in the bath. Sometimes I get a bit sleepy because of the nice water temperature and I suppose you can figure out what happens then. I should get a water resistent e-reader.
Thanks, Marga! Yes, a water-resistant e-reader.
Isn't that just the way with weather! I am sorry you didn't sell as much as you'd hoped but better luck, I hope at the Steam Punk Festival. What luck you having such a good canopy and the rain not spoiling the morning!
I just finished reading your post. What a lovely guest room.
The Steam Punk Festival sounds fun and I hope you have better weather this weekend.🙂 So far I've missed out on a few festivals here, but I'm hoping things improve in July.
Many thanks! We'll have to leave really early, but we are looking forward to the steampunk festival.
There is a scientific explanation for why the weekend is rainier than the week, but I forget it. It's good you have such a professional-looking tent.
Thanks, Quercus! We have put a fair amount of money in our display.
You have a good book to sell – it deserves a good display.


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Headline: 5G and other WiFi standards. Have your say.
In an appeal to the European Union, more than 180 scientists and doctors from 36 countries warn about the danger of 5G, which will lead to a massive increase in involuntary exposure to electromagnetic radiation. The scientists urge the EU to follow Resolution 1815 of the Council of Europe, asking for an independent task force to reassess the health effects.
There is no doubt our standards on regulating electromagnetic radiation is out of date. Should be err on the side of caution or full speed ahead?
In total 3 comments were made and 7 votes were cast. So what is the crowd saying?
Char Staples wrote. If 5G is so harmless, then why have the government been so secretive about it?
Total Votes: Agree:3 Disagree:0 Ratio:100% Agree.
Char Staples wrote. If 5G is so safe, why dispense with the due process of showing evidential proof that it is, as well as encouraging and listening to public input/submissions before rolling it out on mass?
Bex Sansom wrote. There are over 1000 peer reviewed studies demonstrating harm to health. These have been conveniently ignored/disregarded/discredited and selective studies are cited instead. 5G is used by the military for crowd control. For the past few years a group of activists in the US have been saying 5G will be used to track people's data and movements. They were called conspiracy theorists. Then covid hit and their suspicions are now fact. You only have to listen to Bill Gates talking about electronic certificates to show immunity to realise it's like a science fiction horror movie.
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MEGADETH's DAVID ELLEFSON releases sample of first fiction novel, 'Rock Star Hitman'
MEGADETH bassist David Ellefson has set a December 18 release date for his first venture as a fiction thriller novelist, "The Sledge Chronicles: Rock Star Hitman". A two-page teaser from the novel is available at this location.
"The Sledge Chronicles: Rock Star Hitman" is the first chapter in the action-packed saga of Sledge; an up-andcoming musician who uncovers a startling quid pro quo: in exchange for achieving his dreams of rock stardom, he must enlist as a ruthless killer for a clandestine agency. Filled with twists, turns, screams, and even laughs, "Rock Star Hitman" is the cult thriller you've been dying for.
Ellefson states: "I've always wanted to write fiction, and 'The Sledge Chronicles' provided the perfect runway to launch the first of a series of books. I figured I may as well start close to home with a rock 'n' roll figure as the lead character and expand the global exploits from there."
David's canon of published literary work includes four non-fiction titles with his two memoirs "More Life With Deth" (Jawbone Press, 2019) and "My Life With Deth" (Howard/Simon & Schuster, 2014), a self-published lyric-and-poem offering "Unsung: Words & Images" (Blurb, 2012) and his debut how-to book "Making Music Your Business… A Guide For Young Musicians" (Hal Leonard, 1997).
Released through The Ellefson Book Co. imprint, "The Sledge Chronicles: Rock Star Hitman" was co-written with musician, filmmaker and actor Drew Fortier, known for his work with such artists as BANG TANGO, Chuck Mosley (FAITH NO MORE) and ZEN FROM MARS. Fortier also released his autobiography, "Dark, Depressing, And Hilarious", in 2018.
Ellefson and Fortier are no strangers to collaboration; on top of their forthcoming novel, they also have an independent found-footage horror film due out in 2021, "Dwellers", which was written and directed by Fortier, who also stars in the movie, with Ellefson producing via his Ellefson Films imprint.
"The Sledge Chronicles: Rock Star Hitman" can be ordered here.
NO COVER by ELLEFSON
MARATON release live stream video for "THE MANIFEST CONTENT"
BEWITCHER announce "Cursed Be Thy Kingdom" album


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A few passing clouds, otherwise generally clear. Low near 25F. Winds NE at 5 to 10 mph..
A few passing clouds, otherwise generally clear. Low near 25F. Winds NE at 5 to 10 mph.
East Forsyth's Jamison Warren breaks into Ragsdale's secondary.
Woody Marshall/News & Record
East Forsyth's Jamison Warren hauls in a catch for a touchdown in front of Ragsdale's Michael Batts during Friday's NCHSAA Class 4-A West quarterfinal playoff game.
East Forsyth's Chris Chaplin celebrates a touchdown on Friday against Ragsdale.
East Forsyth moves on to the Class 4-A West Region final, defeating Ragsdale in the third round.
By Patrick Ferlise Winston-Salem Journal
Patrick Ferlise
KERNERSVILLE — Familiar territory lies ahead for the East Forsyth football team, now another round deeper in the postseason.
The No. 2 Eagles defeated No. 3 Jamestown Ragsdale 34-21 on Friday night at Fred E. Lewis Stadium in the third round of the NCHSAA Class 4-A playoffs. An offensive push, beginning in the second quarter with 20 points scored, got East Forsyth rolling. Throughout the game, the defense racked up five sacks on Alston Hooker, the Tigers' senior quarterback, as well.
It all led the Eagles (11-2), the defending NCHSAA Class 4-A state champion, back to this point. East Forsyth will appear in its second straight Class 4-A West Region final on Dec. 6 — a road game against No. 1 Greensboro Grimsley at Jamieson Stadium.
Grimsley, which defeated Glenn 32-8 on Friday in their third-round home game, has just one loss this season; it came on Oct. 4 against East Forsyth.
East Forsyth's road to get to a regional appearance again had its challenges, according to Coach Todd Willert. The Eagles faced their fair share of injuries, including junior Jaden Lindsay, a 6-foot-3 and 270-pound offensive lineman, was missing from that win over the Tigers (6-7) because of a knee injury.
"I couldn't be more proud of my coaching staff and my players," Willert said. "You know, right now we've got more than 12 guys out. We're shuffling guys everywhere.
"So, to do what we're doing right now, I'm just excited right now and proud of our kids and coaches that we can fight through this."
East Forsyth trailed 7-0 after the first quarter, but it scored on three straight drives to end the first half.
That began with a 54-yard pass from quarterback Ty Lyles to Jamison Warren for a touchdown as 9:31 remained. Chris Chaplin and Joseph Brown followed it up, scoring the team's two rushing touchdowns. A three-yard run from Brown with 2:24 left pushed the Eagles ahead 20-7. Chaplin ended the night with 63 yards rushing.
Lyles, a junior, passed for 169 yards and three touchdowns. The last came on a 24-yard throw to Jalen Thorns with 1:19 left to seal the win. He made the catch, despite the pass being tipped by Zaymen Taylor, a senior defensive back, in the end zone.
"Definitely the O-line was giving me a lot of time," Lyles said. "And the receivers just found a way to get open. I just had to put (the pass) on their chest."
Despite a second-half surge from Ragsdale, East Forsyth's defense held Devan Boykin to 57 yards rushing. The N.C. State commit scored once off a 39-yard reception with 1:23 left in the third quarter to trail just 28-21 before Thorns' touchdown.
Hooker ended the night with 182 yards in the air. Zyun Reeves, a junior defensive end, helped wrap up Hooker on four of those five sacks.
"We knew coming in the game they had a mobile quarterback and that he could throw the deep ball," Reeves said. "So we prepared in practice, and came in with a game plan knowing we were going to have to do a couple stunts.
"It just started clicking for us."
East Forsyth 34 Ragsdale 21
Ragsdale;7;0;14;0;—;21
East Forsyth;0;20;8;6;—;34
Records: East Forsyth (11-2), Ragsdale (6-7)
RHS — Tyrell Carmichael 5 pass from Alston Hooker (Adnane Choudani kick)
EF — Jamison Warren 54 pass from Ty Lyles (Andrew Conrad kick)
EF — Chris Chaplin 9 run (Conrad kick)
EF — Joseph Brown 3 run (Kick failed)
RHS — Hooker 5 run (Choudani kick)
EF — Joe Ritchens 21 pass from Lyles (Conversion good)
RHS — Devan Boykin 39 pass from Hooker (Choudani kick)
EF — Jalen Thorns 24 pass from Lyles (Conversion failed)
East Forsyth's Elijah Thomas carries the ball against Ragsdale in the first half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Tyrell Carmichael coughs up the ball after being hit by East Forsyth's Isaiah Crowell in the first half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019. Ragsdale recovered the ball.
East Forsyth's Chris Chaplin runs for a touchdown against Ragsdale in the first half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019.
East Forsyth's Chris Chaplin celebrates a touchdown against Ragsdale in the first half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019.
East Forsyth's Jamison Warren makes a touchdown reception as Ragsdale's Michael Batts defends in the first half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Josiah Muldowney runs after a reception in the first half of the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
East Forsyth's Ty Lyles is tackled by Ragsdale's Donavon Platt in the first half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Devan Boykin looks for room to run in the first half of the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's head coach Johnny Boykin talks with Zaymen Taylor during the first half of the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
East Forsyth takes the field for their game against Ragsdale in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Donavon Platt avoids a tackle during the first half of the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
East Forsyth's Jamison Warren breaks into Ragsdale's secondary in the first half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Isaiah Carbajal is tackled by East Forsyth's Morgan Childs during the second half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's head coach Johnny Boykin yells instructions from the sidelines during the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Devan Boykin celebrates his touchdown with teammates during the second half of the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Graham Leslie gets wrapped up by an East Forsyth defender during the second half of the game in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Alston Hooker looks at the scoreboard late in the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's football team stands for the National Anthem before the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
East Forsyth's Ty Lyles tries to slip past Ragsdale's Michael Batts in Kernersville, N.C., on Friday, Nov. 29, 2019.
Ragsdale's Zaymen Taylor encourages his teammates before the game against East Forsyth in Kernersville, N.C., on Friday, Nov. 29, 2019.
East Forsyth celebrates in front of the band after defeating Ragsdale 34-21 in Kernersville, N.C., on Friday, Nov. 29, 2019.
XX HIDE VERTICAL GALLERY ASSET TITLES –
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text:
\section{Introduction}
\label{sec:intro}

Interacting particle systems, or more generally, dynamical systems on graphs/networks, form
one of the major building blocks of modern science~\cite{WattsStrogatz,BarabasiAlbert}. Within
the last three decades, they have permeated virtually all disciplines, ranging from
molecular scales~\cite{Feinberg} to neuroscience~\cite{Sporns}, systems biology~\cite{Palsson},
machine learning~\cite{Haykin}, social science~\cite{Scott2},
epidemiology~\cite{BarratBarthelemyVespignani}, and transportation
networks~\cite{RanBoyce} up to climate science scales~\cite{Dongesetal}. For all-to-all coupled
systems, lattice systems, and various special classes of network dynamics with similar dynamics
at each vertex, there is a well-developed theory to pass to a mean-field
approximation in many theoretical frameworks~\cite{Golse,KissMillerSimon,Kuramoto,LasryLions,
Meleard,Spohn1}. One considers the limit of an infinite graph~\cite{vanderHofstad}
\benn
n\ra \I
\eenn
so the particle/vertex number tends to infinity. If one is interested in the probability
density $\rho=\rho(u,t)$ to find a vertex in a certain dynamical state $u\in\R^k$ at time
$t$, one may often derive a differential equation for $\rho$. One common form for $k=1$
is
\be
\label{eq:VFPEintro}
\partial_t = - \partial_u (\rho V(\rho)),
\ee
where the map $V$ can be derived from the dynamics of each vertex~\cite{Golse,Frank}. If each
vertex also is influenced by Gaussian noise, a second-order term $\partial_{uu}(\cdot)$
commonly appears as well in~\eqref{eq:VFPEintro}. The equation~\eqref{eq:VFPEintro} is sometimes
referred to as Liouville equation, continuity equation~\cite{Strogatz1}, and/or Vlasov equation,
while the second-order equation is more known as Fokker-Planck and/or Kolmogorov equation. Here
we shall refer to this class of equations as Vlasov-Fokker-Planck equation (VFPEs).\medskip

Now it is evident that one would like to derive a far more general form
of~\eqref{eq:VFPEintro}, which also takes into account cases, where the adjacency
matrix $A=A^{(n)}\in\R^{n\times n}$ of the interaction between finitely many vertices
is not just the the matrix of a full graph nor a highly symmetric structure such as a lattice.
There have been many recent steps forward to achieve this goal. Several mathematical
approaches have been successful providing rigorous proofs for VFPEs, where nonlocal
integral terms appear to take into account the heterogeneous coupling
structure~\cite{ChibaMedvedev,LasryLions}.
However, a general theory, which would allow us to upgrade easily from particular cases or
standard all-to-all mean-field limit VFPEs, to modern complex network structures is still
lacking.\medskip

In this note, we propose that a path to achieve an elegant extension of
all classical approaches is to incorporate an analytical approach to graph limit theory.
Graph limit theory~\cite{Lovasz}, i.e., taking $n\ra \I$, has made
significant progress in recent years. New limit structures for dense graphs via
graphons~\cite{LovaszSzegedy}, as well as for sparse
graphs~\cite{BollobasRiordan,Borgsetal,HatamiLovaszSzegedy}, have recently
appeared. The theory is relatively technical and convergence notions are often
hard to understand as they have relied on combinatorial structures~\cite{Lovasz}. Recently a
new approach to unify and extend graph limit theory was proposed by Backhausz and
Szegedy~\cite{BackhauszSzegedy}. Their idea relies on viewing the adjacency matrix
$A^{(n)}$ from an operator-theoretic perspective. In the simplest setting,
one takes a vector $v$ in $\R^n$ and consider the $2\times n$ matrix formed from
$v$ and $v^\top A^{(n)}$. Sampling the columns of this matrix uniformly at random
generates a measure $\mu_v$. Hence, one may view graphs also via their associated
measures. Different graphs, even of different sizes, can then be more easily
compared as we only view them through their \emph{action}; see Section~\ref{sec:graphops}
for more details and a brief review of the class of graphops (graph operators), which
we shall rely on. Backhausz and Szegedy show that in a suitably chosen metric, large
classes of graphs do have non-trivial subsequential limits
\benn
A^{(n)}\ra A^{(\I)}\qquad \text{as $n\ra \I$},
\eenn
where $A^{(\I)}$ can also be viewed as an operator, but now acting as an abstract
operator on a function space. We are going to show (formally) that exactly this
viewpoint is the missing ingredient to start a general theory of VFPEs, and related
classes of evolution equations, as limits of finite-dimensional network dynamics.\medskip

We are going to illustrate our reasoning primarily on a benchmark class of
interacting particle systems, the famous Kuramoto model of coupled
oscillators~\cite{Kuramoto,Strogatz1}. This model is widely studied and has served as
a benchmark case to understand self-organizing and adaptive nonlinear synchronization.
It can be embedded into a more general class of kinetic-type models. We shall see, how these
kinetic models also fit elegantly together with graphops. This approach really seems to
break a barrier that has held back the application of tools from dynamical
systems, functional analysis, and evolution equations to broad classes of
large-scale network dynamics problems.\medskip

The paper is structured as follows: In Section~\ref{sec:graphops}, we review the
relevant parts of graphop theory. In Section~\ref{sec:micro}, we show how this
theory also provides an elegant framework to deal with network dynamics on a
finite-dimensional level, i.e., for finite graphs. For example, we show that
the graphop viewpoint provides a completely natural way to intertwine in one set
of equations the dynamics on the network and the correlation structure between
network elements. Then we proceed to formally develop, why graphops do generalize
previous approaches to VFPEs in Section~\ref{sec:meso}. We conclude in
Section~\ref{sec:outlook} with an outlook towards future opportunities and
challenges.

\section{Graphs as Operators}
\label{sec:graphops}

We build upon the theory of graphops from~\cite{BackhauszSzegedy}. Let $(\Omega,\cF,\mu)$
be a probability space and let $L^p(\Omega,\R)=L^p(\Omega)$ denote the usual Lebesgue
space for $p\in [1,\I]$. A linear operator $A:L^\I(\Omega)\ra L^1(\Omega)$ is called
\emph{P-operator} if the operator norm
\benn
\|A\|_{\infty \ra 1}:=\sup_{v\in L^\I(\Omega)}\frac{\|Av\|_1}{\|v\|_\I}
\eenn
is finite. Key examples of P-operators are matrices, which we want to view as adjacency
matrices of graphs. Indeed, consider $\Omega=\{1,2,\ldots,n\}=:[n]$
for $n\in\N$ with the uniform measure $\mu_{[n]}$ and sigma-algebra given by the
power set $2^{[n]}$. Then a vector $v\in\R^n$ defines a function $v:[n]\ra \R$
by $v(j)=v_j$ for $j\in[n]$. On $([n],2^{[n]},\mu_{[n]})$ any matrix $A\in\R^{n\times n}$
is a P-operator acting on (row) vectors via the usual rule
\benn
(vA)(j)=\sum_{k=1}^n v(k)A_{kj}=\sum_{k=1}^n v_kA_{kj}.
\eenn
Obviously, we can identify $L^\I([n])$ and $L^1([n])$ with $\R^n$ and finite-dimensional
linear operators given by matrices are bounded in the associated operator norm.

A key advantage of general P-operators are their convergence properties, when one
considers sequences of these operators via \emph{profiles}. As an example, we
consider again $A\in\R^{n\times n}$ and any (row) vector $v\in\R^n$, then we have $vA
\in \R^{1\times n}$ so that we may form a matrix $M\in \R^{2\times n}$ with rows $v$
and $vA$. Now we sample columns of $M$ uniformly, which yields a probability measure
$\mu_v$ on $\R^2$. The (1-)profile of $A$ is given by the collection of measures
$\{\mu_v:v\in\R^2\}$. One may generalize the notion of a $1$-profile to a $k$-profile
$\mathcal{S}_k(A)$ for a general P-operator where one uses $k$ vectors $v_1,v_2,\ldots,v_k$
as the first $k$ rows and $v_1 A,v_2 A,\ldots,v_k A$ as $k$ further rows. Hence, the
matrix $M$ becomes a matrix of size $\R^{2k\times n}$; wlog one may restrict
$v_j\in [-1,1]^n$ as the results do not change upon scaling vectors. A random column
in $M$ yields a probability measure on $\R^{2k}$ and $\mathcal{S}_k(A)$ is the collection
of all such measures.

For a general P-operator $A$, consider functions $v_{1},\ldots,v_k,v_1A,\ldots v_kA$
and let
\be
\label{eq:prob1}
\cD(v_1,v_2,\ldots,v_k, v_1A,v_2A,\ldots,v_kA)
\ee
be the joint distribution of the $2k$-tuple, which can be viewed as the pushforward
$(T_{k,A})_*\mu$ of the measure $\mu$ under the map
\benn
T_{k,A}(x)=(v_1(x),v_2(x),\ldots,v_k(x), (v_1A)(x),(v_2A)(x),\ldots,(v_kA)(x)),
\eenn
where $x\in\Omega$. The $k$-profile $\mathcal{S}_k(A)$ is the set of all probability
measures of the form~\eqref{eq:prob1}, where we go through all possible $k$-tuples of
functions in $L^\I(\Omega,[-1,1])$. In summary, the two crucial ideas are to view
more analytically graphs as \emph{operators} and to use profiles to compare
matrices/graphs/P-operators \emph{regardless of their size}. Let
$A,B$ be two P-operators. One defines
a metric
\begin{equation}
d_M(A,B):=\sum_{k=1}^\infty 2^{-k} d_H(\mathcal{S}_k(A),\mathcal{S}_k(B)),
\end{equation}
where $d_H$ is the Hausdorff distance for $X,Y\in\mathcal{P}(R^k)$ (i.e., $X,Y$
are subsets of probability measures on $\R^k$)
\begin{equation}
d_H(X,Y):=\max\left\{ \sup_{x\in X}\inf_{y\in Y} d_{LP}(x,y),
\sup_{y\in Y}\inf_{x\in X} d_{LP}(x,y)\right\}
\end{equation}
and $d_{LP}$ is the L\'evy-Prokhorov metric. This metric is actually quite
natural and given for $\nu_1,\nu_2\in\R^{k}$ by
\begin{equation}
d_{LP}(\nu_1,\nu_2):=\inf\{\varepsilon>0:\nu_1(U)\leq \nu_2(U^\varepsilon)
+\varepsilon\text{ and }\nu_2(U)\leq \nu_1(U^\varepsilon)+\varepsilon\text{
$\forall U\in\mathcal{B}_k$}\},
\end{equation}
where $\mathcal{B}_k$ is the Borel sigma-algebra on $\R^k$ and $U^\varepsilon$
is the set of points having distance less than $\varepsilon$ from $U$. It is
well-known that convergence in the L\'evy-Prokhorov metric essentially is equivalent
to weak convergence of measures so to actually ensure the existence of non-trivial
limits, it is well-balanced choice between retaining some complexity in the limit,
yet still having some form of mean-field object.\medskip

The definition of P-operators generalizes in a natural way to
$\|A\|_{p\rightarrow q}$. One may metrize the space of P-operators via
\be
\label{eq:Popmet}
d_M(A,B):=\sum_{k=1}^\infty 2^{-k} d_H(\mathcal{S}_k(A),\mathcal{S}_k(B)),
\ee
and call the associated convergence notion \emph{action convergence}. Then one can
prove~\cite{BackhauszSzegedy} that any sequence $\{A^(j)\}_{j=1}^\infty$ with uniformly
bounded norm $\|\cdot\|_{p\rightarrow q}$ for $p\in[1,\I)$ and $q\in[1,\I]$ has a
limit~\cite[Thm.~2.14]{BackhauszSzegedy}. This convergence is seen to generalize the
theory of graphons and even graphings (classical objects used for very sparse graphs)
as well as many intermediate cases. For example, if we have a graphon~\cite{Lovasz}
$W:\Omega\times \Omega\rightarrow \R$ with finite norm
\be
\label{eq:graphon}
\|W\|_p:=\int_{\Omega\times \Omega}W(x,y)~\textnormal{d} \mu^2,
\quad p\in[1,\infty),~q:=\frac{p}{p-1},
\ee
then we can define an associated P-operator by
\be
\label{eq:graphonPop}
(fA_W)(x):=\int_{\Omega} W(y,x)f(y)~\textnormal{d}\mu.
\ee
It is easy to see that the norm $\|A_W\|_{p\rightarrow q}$ is finite in this case
and $A_W$ is just an operator-theoretic viewpoint on the $L^p$-graphon $W$. Again,
compactness results hold for the case of $\|\cdot\|_{p\rightarrow q}$ so that we
get the existence of a limiting graphon/P-operator.\medskip

There are also special classes of P-operators, which are particularly
relevant. One important class are \emph{graphops}, which are positivity
preserving and self-adjoint P-operators, which behave effectively like
undirected graphs. This makes sense since in the finite-dimensional case, adjacency
matrices of undirected graphs are \emph{graphops}.

Once we have graphops, it is natural to ask, how these operators are going to
appear in dynamical systems on networks. It is the main theme of this paper to
understand their appearance in dynamics on a formal level (rigorous proofs seem
to be out of reach in full generality at this point but it is expected that
eventually such proofs will be available).

\section{Microscopic Network Dynamics}
\label{sec:micro}

We always denote the dynamical state of a vertex by $u_k(t)$ for $k\in[n]$ and denote the
current time by $t\in[0,T)$ with some fixed $T>0$.

\subsection{The Kuramoto Model}
\label{ssec:Kura1}

The classical Kuramoto model considers oscillators on a circle so
$u_k(t)\in \mathbb{S}^1:=\R/(0\sim 2\pi)$. The dynamics is given by
\be
\label{eq:Kuraclassical}
\frac{\txtd u_k}{\txtd t}=:u'_k=\omega_k+\frac{p_0}{n}\sum_{j=0}^n \sin(u_j-u_k),
\ee
where $\omega_k$ are given internal frequencies of the individual oscillators
and $p_0\geq 0$ is a parameter controlling the coupling. The Kuramoto model on
complex networks is usually written as follows~\cite{Rodriguesetal2}
\be
\label{eq:KuraComplex}
u'_k=\omega_k+p\sum_{j=0}^n A_{kj}\sin(u_j-u_k),
\ee
where $p\geq 0$ is a parameter controlling the coupling, which may contain
a certain scaling in $n$ depending upon the type of graph defined by the adjacency
matrix $A=(A_{kj})_{k,j\in[n]}$. Of course, on a finite-dimensional level the
interpretation of adjacency matrices as graphops on $([n],2^{[n]},\mu_{[n]})$, as
discussed in Section~\ref{sec:graphops}, applies.
So if we consider a matrix of phase differences
\benn
v_{jk}:=\sin(u_j-u_k),\qquad V:=(v_{jk})_{j,k\in[n]},
\eenn
this yields upon inserting it into the Kuramoto model
\be
\label{eq:KuraComplex1}
u_k'=\omega_k+p~\textnormal{diag}(AV),
\ee
where $\textnormal{diag}(M)$ is the vector obtained from the elements on the
diagonal of the matrix $M$. Note that $(V,AV)$ contains the same information as
\benn
(v_{\cdot 1}^\top,v_{\cdot 2}^\top,\ldots,v_{\cdot n}^\top,
v_{\cdot 1}^\top A,v_{\cdot 2}^\top A,\ldots,v_{\cdot n}^\top A)
\eenn
where we used $A=A^\top$ as our graphs are undirected. Hence, $(V,AV)$ can be
viewed as an $n$-profile if we uniformly sample from
it. Since the matrix of phase differences $V$ does \emph{evolve in time},
we see that the input to the dynamics of each oscillator is (in addition
to its intrinsic frequency) driven by moving through a subset of the
$n$-profiles. Even beyond identifying the driving, we can go further by setting
\benn
\omega=(\omega_1,\ldots,\omega_n),\quad u=(u_1,\ldots,u_n),\quad
d=((AV)_{11},\ldots,(AV)_{nn}),
\eenn
to re-write our differential equations as $u'=\omega+pd$, so right-multiplication
and time-differentiation of the $1$-profile $(u,uA)$ gives
\benn
(u,(uA))=(u',(uA)')=(\omega+pd,\omega A,+pdA).
\eenn
Therefore, this defines an evolution equation for the pair $(u,uA)$,
which yields an evolution equation on the space of \emph{measures} $(T_{1,A})_*\mu$
also written by
\be
\label{eq:jed}
\cD(u,uA)=\frac1n\sum_{j=1}^n \delta_{u_j,(uA)_j}
\ee
which are associated to a $1$-profile. In summary, the dynamics of the Kuramoto
model on finite graphs induces an evolution equation on measures, which is somewhat
different from using the classical empirical measure
\benn
\frac1n\sum_{j=1}^n \delta_{u_j},
\eenn
which keeps only track of the positions. Indeed, the $1$-profile also contains the
action of $A$ on $u$ given by $uA$ and thereby the relevant \emph{correlation structure}
embodied by the \emph{joint empirical
distribution} (aka the``JEDi'') given by~\eqref{eq:jed}. The importance of this
novel viewpoint of the dynamics is that once we are on a graph, then we have to
intertwine structure and dynamics~\cite{KuehnNetworks1}, thereby capturing how
vertices respond to connectivity. We will see that this theme re-appears on other
modelling scales and for VFPEs below.

\subsection{The Cucker-Smale Model}

The Cucker-Smale model~\cite{CuckerSmale1} is one well-known model for swarming. One
variant of it reads
\benn
u_k''=\frac1n \sum_{j=1}^n (u_j'-u_k') \psi(|u_k-u_j|),
\eenn
where $u_k=u_k(t)$ is the position of agent $k$ at time $t$ and $\psi$ is a
given function regulating the type of communication between different agents.
More generally, one may pose the model on a network via
\benn
u_k''=\frac1n \sum_{j=1}^n a_{kj}(u_j'-u_k') \psi(|u_k-u_j|).
\eenn
We set $\tilde{a}_{kj}:=a_{kj}\psi(|u_k-u_j|)$, and obtain by re-writing the
model in vectorized form as a first-order system and by right-multiplication with
$A$, the following set of equations
\be
\begin{array}{lcl}
u'&=&v,\\
v'&=& \frac1n\left[\tilde{A}v-\textnormal{Diag}(\tilde{A}\tilde{A}^\top)u\right],\\
(uA)'&=&vA,\\
(vA)'&=& \frac1n\left[\tilde{A}(vA)-\textnormal{Diag}(\tilde{A}\tilde{A}^\top)(uA)\right],
\end{array}
\ee
where $\textnormal{Diag}(M)$ denotes the matrix obtained from $M$ by only keeping the
entries on the diagonal and setting all other entries to zero.
Of course, the equations for $(uA,vA)$ still depend directly on values of $u$ via the
definition of $\tilde{A}$. It is interesting to see that we effectively obtain now
an evolution of measures via the $2$-profile $(u,v,uA,vA)$. The structure of the equations
entails that the second-order nature of the model transfers to profiles, i.e.,
there is a constraint on the $2$-profile in the same sense as for usual second-order
ODEs.

\subsection{Kinetic Models}
\label{ssec:kinbase}

Instead of particular models, one can also look at broader classes of interacting
systems frequently employed in kinetic theory~\cite{Cercignani,Golse,Spohn1}. A form
commonly found in the literature is:
\be
\label{eq:kingen}
u_k'=\sum_{j\neq k} f(u_j,u_k).
\ee
Evidently, one can extend this to a model with a complex network structure by writing
\be
\label{eq:kingen1}
u_k'=\sum_{j=1}^n a_{kj}f(u_j,u_k).
\ee
In this general abstract form, if we set $f(u_j,u_k)=:f_{jk}=f_{jk}(u)$ we get with a matrix
$F(u)=(f_{jk}(u))$ that
\benn
u'=\textnormal{diag}(AF(u))\qquad\text{and}\qquad (uA)'=\textnormal{diag}(AF(u))A,
\eenn
as the evolution equation via the $1$-profile $(u,uA)$. Of course, this formulation is far
too general to see any special structure regarding the evolution equation induced on the measures
contained in profiles. If we linearize the evolution equations around a steady state $u_*$ we obtain
\benn
U'=\txtD_u\left[\textnormal{diag}AF(u)\right]_{u=u_*} U \quad\text{and}\quad
(UA)'=\txtD_u\left[\textnormal{diag}AF(u)\right]_{u=u_*}UA.
\eenn
So locally near steady states the evolution equation for the $1$-profile are identical for
both components, regardless of the precise underlying kinetic/particle model. This already
hints at the fact, that profiles and the operator-theoretic viewpoint via graphops is
well-suited for nonlinear dynamics and functional analysis techniques. This viewpoint will
re-appear again for VFPEs below.

\section{Mesoscopic Network Dynamics}
\label{sec:meso}

Having observed that graphops and profiles can provide a very elegant re-interpretation as
well as augmentation of finite-dimensional network dynamics models, we now proceed to formally
take limits $n\ra \I$ to obtain ``mean-field'' models. We consider the mesoscale, in the
sense that we are no longer interested in the state of individual vertices on a graph but only
in the probability distribution of the state of a vertex.

\subsection{The Kuramoto Model}
\label{ssec:Kura2}

We have already seen in Section~\ref{ssec:Kura1} that the classical Kuramoto model provides a good
starting point. It is well-studied, and its homogeneous mean-field limiting equation is
well-known. We recall its classical formal derivation~\cite{Kuramoto,Strogatz1} here because this
derivation will be important for us to reflect back upon in the context of graphops below.

\subsubsection{All-to-All Coupling}
\label{sec:alltoall}

Starting from~\eqref{eq:Kuraclassical}, suppose we have very large number of oscillators and their
intrinsic frequencies are distributed according to a density $g=g(\omega)$ with
$g(\omega)=g(-\omega)$. One introduces the complex order parameter as
\be
\label{eq:orderpara}
r\txte^{\txti \psi} := \frac1n \sum_{j=1}^n \txte^{\txti u_j}.
\ee
Multiplying the order parameter by $\txte^{\txti u_k}$, and taking imaginary parts, one
easily checks that now the Kuramoto model~\eqref{eq:Kuraclassical} can be re-written as
\be
\label{eq:onefeelsall}
u_k'=\omega_k+p_0r\sin(\psi-u_k).
\ee
In particular, the $k$-th oscillator feels all other oscillators via a single mean-field
parameter. Next, let
\benn
\rho(u,t,\omega)~\txtd u
\eenn
denote the fraction of oscillators with frequency $\omega$ between $u$ and $u+\txtd u$ at time $t$.
Then by construction $\rho$ is a probability density
\benn
\int_{0}^{2\pi} \rho(u,t,\omega)~\txtd u = 1,\qquad \rho\geq 0.
\eenn
In the limit $n\ra \I$, we can formally re-write the order parameter~\eqref{eq:orderpara} as
\be
\label{eq:opcont}
r(t)\txte^{\txti \psi(t)} = \int_0^{2\pi} \int_\R \txte^{\txti u}
\rho(u,t,\omega)g(\omega)~\txtd \omega~ \txtd u.
\ee
Indeed, the last formula can be derived as a limit of the sum in~\eqref{eq:orderpara} using
the law of large numbers~\cite{Strogatz1}. Yet, it is very crucial to note that this law
of large numbers argument only gives us a mean-field as it produces only the mean of the
order parameter completely discarding correlations. Furthermore, the approach implicitly
assumed that each random variable has finite variance. Yet, if these assumptions hold
then we know that the resulting Liouville/continuity equation for the probability density
$\rho$ for the Kuramoto model should be given by
\benn
\partial_t \rho = -\partial_u (\rho v),\qquad v(u,t,\omega):=\omega + p_0 r \sin(\psi-u).
\eenn
This equation can be re-written using the order parameter~\eqref{eq:opcont}, multiplying by a
suitable exponential and taking the imaginary part as
\be
\label{eq:mfKura1}
\partial_t \rho = -\partial_u \left(\rho\left(\omega + p_0 \int_0^{2\pi} \int_\R \sin(\tilde{u}-u)
\rho(\tilde{u},t,\tilde{\omega})g(\tilde{\omega})\txtd \tilde{\omega} \txtd \tilde{u}\right) \right).
\ee
The previous argumentation can be made fully rigorous to derive the mean-field Vlasov-type
equation~\eqref{eq:mfKura1}. Yet, the argument is highly non-trivial and the currently most
elegant way of proof proceeds via a so-called Dobrushin-type bound~\cite{Dobrushin,Dobrushin1},
which is effectively a Gronwall estimate on a metric space of measures, usually carried out in
a Wasserstein metric~\cite{Golse,KuehnBook1,Neunzert}. We remark that if one would consider the
Kuramoto model with stochastic perturbations, then one is going to obtain a (nonlinear)
Fokker-Planck equation~\cite{Frank}, which appears in many classical stochastic coupled oscillator
models, where the coupling appears through a mean field, such as the Desai-Zwanzig
model~\cite{DesaiZwanzig}. Hence, one sometimes refers to~\eqref{eq:mfKura1} as a
Vlasov-Fokker-Planck (VFPE) equation.

\subsubsection{Complex Network Heuristics}
\label{sec:heuristics}

Having dealt with the classical all-to-all coupled VFPE case, it is natural to think about
extensions to treat the Kuramoto model~\eqref{eq:KuraComplex} on complex networks. Recall
that we have assumed that the underlying graph is undirected and unweighted so that the
adjacency matrix $A=(A_{ij})_{i,j\in[n]}$ is symmetric and binary $A_{ij}\in\{0,1\}$ for all
$i,j\in[n]$. There exists a known formal approach~\cite{Rodriguesetal2} to try to ``save'' the
classical mean-field argument from Section~\ref{sec:alltoall}. One defines a local order parameter
as
\benn
r_k\txte^{\txti \psi_k}:=\sum_{j=1}^n A_{kj}\txte^{\txti u_j}.
\eenn
If the graph is sufficiently well-connected and effectively still behaves like an all-to-all
coupled system, one is then tempted to make the ansatz of a single global field
\benn
r\txte^{\txti \psi}=\frac{1}{\kappa_k} r_k \txte^{\txti \psi_k},
\eenn
where $\kappa_k$ is the degree of vertex $k$, i.e., the global field is locally proportional
to the local field weighted by the degree. With this assumption one obtains
that~\eqref{eq:onefeelsall} can now be written as
\benn
u_k'=\omega_k+p_0r\kappa_k\sin(\psi-u_k).
\eenn
In particular, these steps motivate that one might want to use the local degree of a vertex
as the next approximation criterion to derive a mean field~\cite{Ichinomiya}. Hence, one
option is to consider a probability density $\rho(u,t,\omega,\kappa)$ of oscillators having
phase $u$, frequency $\omega$, and degree $\kappa$ at time $t$ with the usual conditions
\benn
\int_{0}^{2\pi} \rho(u,t,\omega,\kappa)~\txtd u = 1,\qquad \rho\geq 0.
\eenn
Evidently one cannot really do much with standard tools as just defining $\rho$ still does
not lead to a mean-field VFPE. If one assumes that the network is \emph{uncorrelated} and
has a degree distribution $d(\kappa)$, then the probability that an edge has its end a
vertex of phase $u$, degree $\kappa$ and frequency $\omega$ at time $t$ is
\benn
\frac{\kappa d(\kappa)}{\langle \kappa \rangle} g(\omega) \rho(u,t,\omega,\kappa)
\eenn
where $\langle \kappa\rangle$ is the average degree of a vertex in the graph.
This suggests that it might be helpful to define the order parameter now as
\benn
r\txte^{\txti \psi}:=\int_0^{2\pi}\int_\R\int_0^\I
\frac{\kappa d(\kappa)}{\langle \kappa \rangle} g(\omega)
\rho(u,t,\omega,\kappa)~\txtd \omega~\txtd \kappa ~\txtd u.
\eenn
Now the same trick as previously, multiplying by an exponential and taking
imaginary parts, yields VFPE equation
\benn
\partial_t \rho = -\partial_u \left(\rho (\omega + p \kappa r\sin(\psi-u))\right).
\eenn
The right-hand side of the VFPE could again be expressed now as
some integral. Indeed, one
may formally write the equation for the evolution of the average phase in the limit
$n\ra \I$ as
\be
u'=\omega+\frac{p\kappa}{\langle \kappa \rangle}\int_0^{2\pi}\int_\R\int_0^\I
g(\tilde{\omega})d(\tilde{\kappa})\tilde{\kappa}\rho(\tilde{u},t,\tilde{\omega},\tilde{\kappa})
\sin(\tilde{u}-u) ~\txtd \tilde{\kappa} ~ \txtd \tilde{u} ~\txtd \tilde{\omega},
\ee
which arises as a formal replacement of the sum in the Kuramoto
model~\eqref{eq:KuraComplex}. This suggests a closed VFPE in the form
\be
\label{eq:mfKura2}
\partial_t \rho = -\partial_u \left(\rho \left(
\omega+\frac{p}{\langle \kappa \rangle}\int_0^{2\pi}\int_\R\int_0^\I
g(\tilde{\omega})d(\tilde{\kappa})\tilde{\kappa}\rho(\tilde{u},t,\tilde{\omega},\tilde{\kappa})
\sin(\tilde{u}-u) ~\txtd \tilde{\kappa} ~ \txtd \tilde{u} ~\txtd \tilde{\omega}\right)\right).
\ee
The last considerations already show an emerging theme that we shall exploit later on:
there is an effective integral \emph{operator acting on the density} $\rho$, which depends
upon the graph structure, in the mean-field VFPE equation on complex networks. In
comparison to the classical all-to-all coupled VPDE~\eqref{eq:mfKura1}, we have
replaced in~\eqref{eq:mfKura2}
\benn
\rho(\tilde{u},t,\tilde{\omega}) \qquad \textnormal{by}
\qquad \int_0^\I \frac{d(\tilde{\kappa})\tilde{\kappa}}{\langle
\kappa\rangle} \rho(\tilde{u},t,\tilde{\omega},\tilde{\kappa})~\txtd \tilde{\kappa}.
\eenn

\subsubsection{Kuramoto on Graphons}
\label{sec:graphons}

Regarding the previous considerations in Section~\ref{sec:heuristics} on integral operators,
it is now no surprise, how the VFPE equation on graphons should look~\cite{ChibaMedvedev}.
Let $W:\Omega\times \Omega\ra \R$ be a graphon as defined in~\eqref{eq:graphon} and recall
that it defines an associated P-operator $A_W$ via~\eqref{eq:graphonPop}. For graphons we
know that we may identify $\Omega=[0,1]$ with the unit interval. So for $x\in \Omega$ one
now obtains the VFPE
\be
\label{eq:mfKura3}
\partial_t \rho = -\partial_u \left(\rho \left(
\omega+p\int_0^{2\pi} \int_\R \int_\Omega
g(\tilde{\omega})W(x,\tilde{x})\rho(\tilde{u},t,\tilde{\omega},\tilde{x})
\sin(\tilde{u}-u) ~\txtd \tilde{x} ~ \txtd \tilde{u} ~\txtd \tilde{\omega}\right)\right).
\ee
for a density $\rho=\rho(u,t,\omega,x)$ of oscillators having phase $u$, frequency $\omega$,
and graphon position $x$ at time $t$. The position $x\in [0,1]$, where we evaluate a graphon has
the natural interpretation in the graph limit as the position/index of a vertex, while $W(x,y)$
provides a limiting version of the adjacency matrix, i.e., how $x$ is connected to $y$. In
comparison to the classical all-to-all coupled VPDE~\eqref{eq:mfKura1}, we have
replaced in~\eqref{eq:mfKura3}
\benn
\rho(\tilde{u},t,\tilde{\omega}) \qquad \textnormal{by} \qquad \int_{\Omega} W(x,\tilde{x})
\rho(\tilde{u},t,\tilde{\omega},\tilde{x})~\txtd \tilde{x}.
\eenn
Now the generalization to far more general classes beyond graphons is becoming
physically evident.

\subsubsection{Kuramoto on Graphops}
\label{sssec:dyngraphons}

Let $A:L^p(\Omega)\ra L^q(\Omega)$ be a graphop as defined in Section~\ref{sec:graphops}.
Suppose it arises as a limit from a sequence of adjacency matrices
\benn
A=A^{(\I)}=\lim_{n\ra \I} A^{(n)},
\eenn
where convergence is understood with respect to the metric $d_M$. From the Kuramoto model
on complex networks~\eqref{eq:KuraComplex} one now formally obtains the VFPE
\be
\label{eq:mfKura4}
\partial_t \rho = -\partial_u \left(\rho \left(
\omega+p\int_0^{2\pi} \int_\R g(\tilde{\omega})(A^{(\I)}\rho)(\tilde{u},t,\tilde{\omega},x)
\sin(\tilde{u}-u) ~\txtd \tilde{x} ~ \txtd \tilde{u} ~\txtd \tilde{\omega}\right)\right).
\ee
for a density $\rho=\rho(u,t,\omega,x)$ of oscillators having phase $u$, frequency $\omega$,
and graphop coordinate $x$ at time $t$. The position $x\in \Omega$ is now more abstract, yet
it still has a meaning in the sense that it should present the position/index of a vertex.
Instead of an integral operator representation, for a P-operator, we have to think of the
action of a graph, which is actually the primary insight that recently arose in the graph
theory literature as discussed in Section~\ref{sec:graphops}. In~\eqref{eq:mfKura4}, we see,
how this insight can be carried over to easily writing abstract VFPE mean-field type equations,
even if the graph is neither all-to-all, nor uncorrelated, nor homogeneous, nor dense, nor
a special sparse graph. In comparison to the classical all-to-all coupled VPDE~\eqref{eq:mfKura1},
we have abstractly replaced in~\eqref{eq:mfKura4}
\benn
\rho(\tilde{u},t,\tilde{\omega}) \qquad \textnormal{by} \qquad (A^{(\I)}\rho)(\tilde{u},t,
\tilde{\omega},\tilde{x}).
\eenn
Note that this formulation is much cleaner, more general, and much easier to comprehend in
comparison to other approaches.

\subsection{Kinetic Models on Graphops}
\label{ssec:kingra}

The approach in the previous section for the Kuramoto model is now relatively
easy to \emph{formally} generalize to more abstract kinetic models such as~\eqref{eq:kingen1}.
One simply takes the normal VFPE for~\eqref{eq:kingen}, which would read
\be
\partial_t \rho = -\partial_u (\rho V(\rho)),
\ee
where $V$ is completely computable from $f$~\cite{Golse}. Then one lifts the action of the
finite-dimensional adjacency matrices $A^{(n)}$ onto a limiting graphop $A^{(\I)}$, and
replaces the density in the vector field driven part $V$ of the VFPE
\be
\label{eq:limitfinally}
\partial_t \rho = -\partial_u(\rho V(A^{(\I)}\rho)),\qquad \rho=\rho(u,t,x),~x\in\Omega.
\ee
All the operator-theoretic properties of the finite-dimensional graphs $A^{(n)}$,
which persist in the limit $A^{(\I)}$, actually can now be used to analyze the VFPE. We know
from~\cite{BackhauszSzegedy} that graphops do carry a lot of information via the graph limit
for large classes of graphs, so we may strongly expect that an analysis of~\eqref{eq:limitfinally}
can now proceed along classical lines of dynamical systems theory. For example, steady states
$\rho_*$ will satisfy
\benn
0=\partial_u(\rho_* V(A^{(\I)}\rho_*)).
\eenn
Linearization is possible via the normal Fredholm derivative in many cases. Suppose that
we have an evolution equation
\be
\label{eq:evocool}
\partial_t \rho = -\partial_u(\rho V(A^{(\I)}\rho))\qquad \rho=\rho(t)\in\cX,
\ee
in a suitable Banach space $\cX$ where the P-operator part is such that it defines
a well-defined evolution in $\cX$.
Then the linearization is just formally given by the chain rule and product rule
\benn
\partial_t \Xi = -\partial_u(\Xi V(A^{(\I)}\rho_*) + \rho_* \txtD_\rho V(A^{(\I)}\Xi) )
=:\cL_\I \Xi
\eenn
for $\Xi=\Xi(t)\in\cX$. The linear
operator $\cL_\I$ can now be analyzed using classical tools from functional analysis
such as spectral theory. As in Section~\ref{ssec:kinbase} on the finite-dimensional
level, the operator $\cL_\I$ carries in an intertwined way the information about
\begin{itemize}
 \item the shape of the steady state via $\rho_\I$,
 \item the linearization of the coupling function $f$ via $V'$,
 \item the graph spectral information via $A^{(\I)}$.
\end{itemize}
Hence, the graphop viewpoint provides in a completely natural way the opportunity to
lift spectral information from the finite-dimensional setting to the graph limit in
the context of stability analysis.

\section{Outlook}
\label{sec:outlook}

We have shown that the viewpoint of a more abstract operator-theoretic approach is
extremely elegant to provide a general theory for network dynamics, when we do not
have a simple coupling structure. This applies to the finite-dimensional microscopic
case as well as the limiting case for VFPEs.

Although we have physically derived, by using analogies to previous approaches in the
mean-field context for homogeneous/all-to-all coupling, a suitable VFPE formulation
on a limiting graphop, several questions remain. From a kinetic theory perspective,
we would like to prove that the limiting VFPE indeed approximates over a finite
time scale, the underlying finite-dimensional network dynamics for large $n$. We
conjecture that similar proofs\footnote{A rigorous proof is currently work in
progress.} via a Dobrushin-type estimate can also work in the graphop context,
when the structure of the graph limit theory is taken into account. From the dynamics
perspective, it would be very interesting to study spectral theory for operators
involving graphops as one element of their definition in more detail as we have
indicated in the last section. Furthermore, from the physical perspective it is
important to gain a better understanding of graph limit procedures in the sense
of particle interactions. We believe that the finite-dimensional
observations we provided hint at a correlation structure viewpoint being
particularly important to provide the correct physical interpretation. From the
viewpoint of direct applications in various sciences, it seems reasonable to assume
that testing examples and analyzing finite-size effects are going to be of particular
practical importance.\medskip

\textbf{Acknowledgments:} CK acknowledges partial support of the DFG via the SFB/TR 109
Discretization in Geometry and Dynamics, partial support by the TiPES (Tipping Points in the
Earth System) project funded the European Union's Horizon 2020 research and innovation
programme under grant agreement No 820970 and support by a Lichtenberg Professorship of the
VolkswagenStiftung. CK also thanks the TUM International Graduate School of Science and
Engineering IGSSE for support via the project SEND (Synchronization in Evolutionary Network
Dynamics). CK also appreciated general discussions with Sebastian Throm and Tobias B\"ohle
regarding limits of network dynamics.

{
\small
\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Villanos hace referencia a varios artículos:

Definición
 Personas que viven en las villas;
 Villano, De los Siervos de la Edad Media, surgieron los Villanos Libres de las primeras ciudades; de este estamento urbano salieron los primeros elementos de la Burguesía.

Multimedia
 Villanos (banda), grupo musical argentino formado en 1994
 Villainous, miniserie web de Cartoon Network.

Además
Isla Villano, en la provincia de Vizcaya, España.

redpajama_set_name: RedPajamaWikipedia


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by: Carter - posted: 10/14/2019 5:19:00 PM
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Q: Parse to integer with argp.h I would like to be able to receive program arguments and options in my C program. The options should be treated as floats or ints. For some reason, I couldn't find good articles, tutorials, docs about argp.h. I started my implementation with the examples on the GNU Libc Manual, though unfortunately, it didn't get me to the solution.
Here is how I tried to solve the problem (example can be compiled, included every necessary line):
#include <stdlib.h>
#include <argp.h>

static char doc[] = "Doc";
static char args_doc[] = "ARG1";

static struct argp_option options[] = {
  {"bool", 'b', 0, 0, "Simple boolean flag, works as I expected."},
  {"int", 'i', 0, 0, "Would like to be able to parse options as --int=4 or -i 4."},  // but I can't
  {0}
};

struct arguments {char *args[1]; int xbool, xint;};

static error_t
parse_opt (int key, char *arg, struct argp_state *state) {
    struct arguments *arguments = state->input;
    printf("key = %c, arg = %s\n", key, arg);  // My attempt to understand the problem
    //if (arg == NULL) return 0;  // Prevents segfaults, in order to see how the args and keys change
    switch (key) {
        case 'b': arguments->xbool = 1; break;
        case 'i': arguments->xint = (int) strtol(arg, NULL, 10); break;
        case ARGP_KEY_ARG: if (state->arg_num >= 1) {argp_usage(state);} arguments->args[state->arg_num] = arg; break;
        case ARGP_KEY_END: if (state->arg_num < 1) {argp_usage(state);} break;
        default: return ARGP_ERR_UNKNOWN;
        }
    return 0;
}

static struct argp argp = { options, parse_opt, args_doc, doc };

int main (int argc, char **argv) {
    struct arguments arguments;
    arguments.xbool = 0;
    arguments.xint = 0;
    argp_parse (&argp, argc, argv, 0, 0, &arguments);
    printf("ARG1 = %s\nbool option = %s\nparsed int option = %d",
           arguments.args[0], arguments.xbool ? "true" : "false", arguments.xint);
    exit (0);
}

The simple boolean flag works, as expected, so ./myprogram.out myarg --bool and ./myprogram.out myarg -b changes the flag's value.
However, I can't seem to find a way to parse the arguments as integers or floating point numbers.
This it the output I get for ./a.out -b --int=2 myarg:
key = , arg = (null)
key = b, arg = (null)
./a.out: option '--int' doesn't allow an argument
Try `a.out --help' or `a.out --usage' for more information.

and for ./a.out -b --int 2 myarg I get a segmentation fault, as I try to parse a NULL: key = i, arg = (null). I added a NULL check, and this way I could see, that the option I would like to parse comes with a , key (expected to come with i).
key = i, arg = (null)
key = , arg = 2

I thought about using a library because the program needs to handle various float and int options, falling back to default values, and I've seen often that it's not recommended to roll your own argument and options parser. Based on the examples, argp.h looked promising, however I couldn't get it to work yet.
ps. I know that parsing directly to ints and floats are not part of argp, this is why I was (naively, it seems) trying to add it to the structures and parse_opt function.

A: As it turned out, there was an error with options[]. The third, const char *arg parameter in the argp_option struct must be provided, if the option has argument associated with it [source: GNU C: Argp Option Vectors].
static struct argp_option options[] = {
  {"bool", 'b', 0, 0, "Simple boolean flag, works as I expected."},
  {"int", 'i', "Numbah", 0, "Now everything works as expected, I get the correct key-value (key-arg) pair in the parse_opt function"},
  {0}
};

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