text:
New Suez Canal opening a moment of national pride for Egypt
December 18, 2015 by David Frederick
Strong media relations effort highlights economic importance of $8 billion project
An American aircraft carrier in the Suez Canal (Image via Wikipedia commons).
Client: Suez Canal Authority
Agency: Ogilvy Public Relations
Campaign: New Suez Canal – Egypt's Gift to the World
In early August 2015, Ogilvy Public Relations finessed a major international event as a significant achievement for Egypt and Egyptian President Abdel Fattah el-Sisi – the opening of the new Suez Canal.
Leveraging the expertise of WPP, the Ogilvy team created an integrated communications campaign spanning advertising, PR, social media, and events. A senior global media team was tapped to handle global media relations, as well as targeted in-market support.
Ogilvy helped the new Suez Canal showcase this major achievement for a modern Egypt.
The Suez Canal accounts for 8% of the world's maritime trade and brings in $5.5 billion annually into the Egyptian economy. When Sisi proposed an ambitious plan to expand the canal, it was met with patriotic fervor from the public.
To accommodate modern ships and to adapt to an increase in world trade, Sisi planned to build a 23-mile channel parallel to the original canal: this would allow two-way traffic for the first time and reduce waiting times for vessels by as much as eight hours. The project was fully financed by Egypt, which raised the $8 billion budget in eight days and, despite projections that canal expansion would take three years to build, it was completed inside 12 months.
The aim was to counter arguments from some economists who questioned whether the canal could deliver on its economic goals by leveraging Suez Canal Authority research that showed it would accelerate the Egyptian economy for decades to come.
At the heart of Ogilvy's strategy was the belief that an earned media campaign on a global scale would generate stories with the journalistic credibility and gravity necessary for the canal to earn support from global leaders in trade, politics, and Egypt itself. Ogilvy teams around the world were tasked with bringing journalists to Egypt to report on the opening.
"We targeted the most influential global media, such as CNN, BBC, Wall Street Journal, Financial Times – those outlets that drive conversation globally. We leveraged all of those folks based in Cairo," explains Jennifer Risi, MD of media influence and head of media relations for North America at Ogilvy PR. "The other part of the strategy was in-market activity. We targeted media that had a high profile within the market, as well as a focus on investments. The purpose of the new canal is to open up investments within the region so that this could be seen not only as a symbol of the region, but also as a gateway of investments in the Middle East overall."
Ogilvy activated its global network to reach a worldwide audience with outreach tailored for each region, leveraging teams across the Middle East, North Africa, Europe, Asia, and the Americas to tap local media around the world. Strategic use of pre-interviews with high-level canal personnel led to preview stories in major global papers in the week before the opening.
The agency arranged for more than 100 top-tier journalists to attend the opening ceremonies in Cairo, sitting alongside the Egyptian people and witnessing the first ships pass through the new canal. Arrangements were made for TV news anchors to broadcast live from the banks of the canal and Ogilvy managed an on-site media bullpen in the desert.
"We had to educate a government on how to really manage communications at this level. To do something like this, you're inviting the world into your country to witness this major moment. And we had to make it authentic, give the right access, and show that there is a new modern Egypt, not just in the infrastructure of the canal, but in how we were welcoming to the country," says Risi.
Efforts resulted in more than 5,000 pieces of coverage garnering more than 7.5 billion media impressions. In North America alone, Ogilvy's efforts resulted in more than 800 pieces of global coverage including multiple stories from The New York Times, CNN, Associated Press, CNBC, and CBS News. Exclusive interviews led to full-page stories in The New York Times, USA Today, and The Wall Street Journal.
The day-of launch was attended by several world leaders, including French President François Hollande and Russian Prime Minister Dmitry Medvedev. El-Sisi called the new canal "Egypt's gift to the world" and presided over a parade of naval vessels and a flyover of F-16s.
Lobbying & public affairs
Ogilvy PR hires RF/Binder's Karen Paff for Media Influence team
WPP, Ogilvy PR prep for wave of journalists at second Suez Canal opening
News Analysis: How to turn delay to an advantage


Our family received tickets for a carriage ride around the Gaylord Opryland Hotel.
We were so excited to see the Opryland Hotel's Christmas lights and decorations.
But — as we learned the hard way — the Opryland Hotel is no place to take your family Christmas photo!
The carriage rides take you around the Opryland Hotel's circular driveway, up and down the alley of white-lighted trees.
The carriages are pulled by Tennessee Walking Horses. They are so huge!
After our short ride, we went inside and journeyed from one end of the hotel to the other.
We had to! In order to save the $16 SELF PAY parking fee, we had parked at the Opry Mills mall and walked over to the Opryland Hotel.
Touring the lights — inside and out — is FREE.
Our walk inside the Opryland Hotel started the Magnolia Portico (what used to be the main hotel lobby), with its grand staircase and immense poinsettia tree.
Down the hallway, we detoured through a life-size nativity.
Opryland's Walk Through Bethlehem takes you outdoors in a small loop around a pool of water.
Then we wound through Opryland's conservatories. Both the Cascades and the Delta Island offer lush greenery, fountains, and ropes of lights hanging from the glass ceiling.
Taking a good photo in the semi-darkness there is nearly impossible — at least at night.
During the day, the inside of the Opryland Hotel is filled with natural light, which is much better for taking Christmas photos.
Here are 3 things you absolutely must see during your next visit to the Opryland Hotel.
At night, your best bet for photos will probably be either of the hotel lobby areas.
The newly renovated Cascades Lobby — where the concierge desk is located — has impressive columns and fun figures like this elf.
The Cascades Lobby is where you go to buy tickets to Opryland Hotel events like the Radio City Christmas Spectacular, the Pam Tillis Dinner Show, Grinch On Ice, and other seasonal shows.
Again, even the white-marbled Cascade Lobby at night is lit with pockets of dim or harsh lighting.
That didn't stop us from trying, though!
After all our walking, I think my favorite photos were taken in the main Magnolia Lobby.
The deep reds of the poinsettia tree, the grand staircase and even the richly patterned carpet make the best backdrop — provided you aren't taking photos of a toddler!
By the way, if you're never been, Opryland's Christmas lighting ceremony is rather Disney-like.


text:
'Every Last Child' campaign to reach two million 'invisible' children
Wednesday | 27th April 2016
New Delhi, 26 April. Save the Children today launched its new global campaign, entitled 'Every Last Child' at an event held here. The campaign aims to improve the lives of two million of India's most excluded and vulnerable children, including 500,000 children currently living on the streets. Save the Children International's CEO, Helle Thorning-Schmidt (former Prime Minister of Denmark), was joined by Venkaiah Naidu (Minister of Urban Development, Housing and Urban Poverty Alleviation and Parliamentary Affairs), as well as Stuti Kacker (Chairperson, NCPCR), Swaroop Sampat (Actor and Educationist) and Sonia Singh (Editorial Director, NDTV), Harpal Singh (Vice Chair, Save the Children International & Chair, Save the Children in India), civil society and UN representatives and over 150 children from the slums of Delhi, to stand up for the rights of these children.
Venkaiah Naidu, Minister of Urban Development, Housing and Urban Poverty Alleviation and Parliamentary Affair said "This is a very important campaign and would like to congratulate Save the Children for working consistently with children. Time has come to imbibe our age old values and ensure cities that are child friendly. Anything good for our children is good for our entire society. This is my firm belief. Millions of children still live on the street who are so vulnerable. Smart cities will have child friendly plans and implementation . Civil society will have to work with the government and that's why this campaign by Save the Children is so important."
The campaign draws on the findings of a DFID-supported research report on street children in India, which includes research from five cities and reveals the severe hardship and risks faced by India's street children. The findings reveal that over a quarter, 26%, of street children are under six years of age and that non metro cities had the highest percentage of homeless children, with 1.3% in Patna and Mughalsarai against 0.4% in Hyderabad and Kolkata.
Millions of 'invisible' children living in slums and on the streets are not registered at birth and do not have identity documents, excluding them from accessing vital services. Save the Children is calling for immediate action to ensure that all children are registered and given access to the critical health, protection and education services they need. These services are especially vital to street children who are among the most at risk of abuse, exploitation and neglect.
Thorning–Schmidt emphasized Save the Children's commitment to reaching the world's most excluded children and said, "Since India has the highest number of children in the world, improvement in their lives will have a huge impact for the entire world."
Outcomes for poor children in urban settings are significantly lower than for higher-income families: a poor child is almost one and half times more likely to die in his/her first month compared to a child born to a higher-income family. 63% of street children were found to be illiterate.
The new research also shows that 85% of street children live with an adult caregiver. The majority had come from poor rural areas only to find themselves homeless in the cities. Evidence suggests that the poor in India are increasingly marginalised as the gap between the richest and the poorest has widened dramatically.
Every Last Child is a three-year campaign calling on decision makers at the household, local, national and international levels to ensure barriers that prevent the poorest children from accessing life-saving services are eliminated.
The organisation also calls on world leaders to commit to three basic guarantees. These include:
Fair financing – so that essential services are available to even the poorest households;
Equal treatment and access to services for all children
Decision makers to be held accountable for ensuring that every attempt is made to reach every last child's needs.
"India and the world have seen enormous gains for children over the last decade," said Harpal Singh, Vice Chair, Save the Children International, adding, "But not every child has benefited. In fact, millions of children have been completely left behind, and are still living in extreme poverty, their life chances blighted. With the government promoting the concept of Smart Cities, an urban revolution can be expected to take place in India. Along with the leaps in technology, it will be imperative to ensure that our urban landscape, new and old, is child-friendly."
Save the Children calls upon the government to issue identity proofs to all the street children in India and provide basic healthcare, education and protection amenities to them.
Save the Children works across 20 states of India and in 120 countries worldwide. It focusses on issues related to education, health, protection and humanitarian/DRR needs of children, especially for those who are the most deprived and marginalized.
Helle Thorning-Schmidt joined Save the Children International as CEO on April 4th 2016.
For further information — including interviews with spokespeople — please contact:
Devendra Tak Mobile: +91 98111 68488 Email: [email protected]
Visit www.savethechildren.in

redpajama_set_name: RedPajamaCommonCrawl


You're invited to visit the many parks, open spaces and trails in Bend. It is a great way to connect with the outdoors and with the community.
Get to know all of Bend's parks and how to visit.
Hit the trail! Discover the miles of trails available for walking, biking and more.
If you'd like to rent out a facility or park area for your event, visit the facilities page.
Learn more about park rules and regulations, including business use of parks, filming in parks and metal detectors.
Each new park, trail and facility is developed by working with the community to meet that particular area's needs. We currently have several park, trail and facility projects under development. To learn more about the status of a particular park project, visit the Current Projects page.
To learn about how we maintain and care for the parks visit the maintenance page. If you see a park maintenance emergency, you can call our 24-hour on call number (541) 410-3319.


text:
BudgIT Employees Receive Serious Threats After Exposing Fashola's Embezzlement Of Public Funds
BudgIT released a statement on Tuesday titled "We refuse to be intimidated" explaining its disappointment that a threats would be made against its staff when it merely published information that was already publicly available.
BudgIT, an organization dedicated to highlighting details of government budgets to the Nigerian public, has received serious threats after exposing that former Governor Babatunde Fashola of Lagos State spent N78.3 million of public funds to develop his personal website, the civil society organization said on Tuesday.
In a conversation with SaharaReporters, BudgIT staff member Stanley Ochonu said that the group received a threat he called both "very serious" and "very, very credible" on Monday, and that he and his colleagues were taking security precautions in response.
"We are being very careful, and I can't go into the specifics until we've spoken to our lawyers," Mr. Ochonu said. He also said that the organization was "very certain" that the threat was related to uncovering the cost of former Governor Fashola's personal website using public funds.
"We reiterate that the information was, and (at the timing of this statement), still is on the website, and therefore in the public domain, and therefore legally open to scrutiny by BudgIT," the statement said. (The information is still available on the Lagos State government's website, and can be found here: http://www.lagosppa.gov.ng/awarded-contracts/details-contracts-award-2014.)
By issuing the press release, Mr. Ochonu said that BudgIT "wanted to make sure that the public is aware that something is going on."
The group also put Inspector General of Police Solomon Arase and Lagos State Commissioner of Police Ajani Owoseni "on notice, in case anything should happen to any member of our staff" or affiliated persons, according to the release.
Corruption CRIME Politics
Corruption CRIME Money Politics
Bayelsa's Governor Dickson Blasts APC Chairman Over N440 Billion Allocation
Corruption CRIME Legal Police Politics
Lagos Banker Reveals Governor Emmanuel Udom Of Akwa Ibom Threatened Her Life Over Financial Fraud
Atiku Names 30 Corrupt People Working With Buhari
Corruption CRIME Legal Politics
Jonathan's Aide, Omokri Burglarizes House In Delta, Carts Away N25Million Meant For Buying Of Votes
Corruption CRIME Legal Politics Scandal
Patience Jonathan Appeals Forfeiture Of $8.4 Million, N9.2 Billion To Nigerian Government
Forgery, Perjury: Anambra Youth Group Demands Andy Uba's Arrest, Prosecution


March is almost here and many of us who made commitments heading into 2017 are starting to evaluate our progress . How are you doing?
Day 31 Bonus: In With The New: What To Do With 2017?
Now that we've had 30 days of challenges for a better 2017, it's time to make the rubber meet the road. In with the new.
Day 30: Out With The Old: What to Do With 2016?
The new year can be a conundrum for many people. Turning over the calendar pressures us all to take a look back and a look forward. The old saying "out with old; in with the new" presents an awkward challenge to look back and look forward. But is the new year really all about resolutions, or is it more about goals?
Doctors have known for a long time that positive values such as generosity, honesty, telling the truth, and sharing have real life health benefits such as lowering blood pressure, living longer, and reducing stress.
Listen. Improve. Repeat. How often do you hear that formula when it comes to marketing your books? Instead, everyday we are bombarded with some new cutting edge formula for selling millions of books on Amazon.


text:
\section*{Introduction}

Many important phenomena that we would like to understand $-$ formation of public opinion, trending topics on social networks, movement of stock markets, development of cancer cells, the outbreak of epidemics, and collective computation in distributed systems $-$ are closely related to predicting large-scale behaviors in networks of locally interacting dynamic agents. Perhaps the most widely studied and mathematically intractable of such collective behavior is the \textit{synchronization} of coupled oscillators (e.g., blinking fireflies, circadian pacemakers, BZ chemical oscillators), and has been an important subject of research in mathematics and various areas of science for decades \cite{strogatz2000kuramoto, acebron2005kuramoto}. Moreover, it is closely related to the \textit{clock synchronization} problem, which is essential in establishing shared notions of time in distributed systems and has enjoyed fruitful applications in many areas including wildfire monitoring, electric power networks, robotic vehicle networks, large-scale information fusion, and wireless sensor networks
 \cite{dorflfer2012synchronization, nair2007stable, pagliari2010scalable}.

For a system of deterministic coupled oscillators (e.g., the Kuramoto model \cite{kuramoto2003chemical}), the entire forward dynamics (i.e., the evolution of phase configurations) is analytically determined by 1) the initial phase configuration and 2) the graph structure (see Figure \ref{fig:dataset_visualization_full}). In this paper, we are concerned with the fundamental problem of \textit{predicting whether a given system of coupled oscillators will eventually synchronize}, \commHL{using some information on the underlying graph or on the initial dynamics (that is, early-stage of the forward dynamics).} More specifically, we consider the following three types of synchronization prediction problems (see Figure \ref{fig:dataset_visualization_training}):
\begin{customquestion}{Q1}\label{Q1}
Given the initial dynamics and graph structure, can we predict whether the system will eventually synchronize?
\end{customquestion}

\begin{customquestion}{Q2}\label{Q2}
Given the initial dynamics and not knowing the graph structure, can we predict whether the system will eventually synchronize?
\end{customquestion}

\begin{customquestion}{Q3}\label{Q3}
Given the initial dynamics partially observed on a subset of nodes and possibly not knowing the graph structure, can we predict whether the whole system will eventually synchronize?
\end{customquestion}

Analytical characterization of synchronization would lead to a perfect algorithm for the synchronization prediction problems above. However, while a number of sufficient conditions \commHL{on graph topology \cite{eom2016concurrent,boccaletti2006complex,chowdhury2020effect, lyu2015synchronization} (e.g., complete graphs or trees)}, model parameters (e.g., large coupling strength) or on initial configuration (e.g., phase concentration into open half-circle) for synchronization are known, obtaining an analytic or asymptotic solution to the prediction question, in general, appears to be out of reach, \commHL{especially when these sufficient conditions for synchronization are not satisfied. Namely, we are interested in predicting the synchronization of coupled oscillators where the underlying graph\commHL{s are non-isomorphic} and the initial configuration is not confined within an open half-circle in the cyclic phase space.}
Since the global behavior of coupled oscillators is built on non-linear local interactions, as the number of nodes increase and the topology of the underlying graphs become more diverse, the behavior of the system becomes rapidly intractable. To provide a sense of the complexity of the problem, note that there are more than $10^{9}$ non-isomorphic connected simple graphs with $11$ nodes \cite{combinatorial_data}.

However, the lack of a general analytical solution does not necessarily preclude the possibility of successful prediction of synchronization. In this work, we propose a radically different approach to this problem that we call \textit{Learning To Predict Synchronization} (L2PSync), where we view the synchronization prediction problem as a binary classification problem for two classes of `synchronizing' and `non-synchronizing', \commHL{based on the fact that any given deterministic coupled oscillator system eventually synchronizes or converges to a non-synchronizing limit cycle.} \commHL{In this work, we consider three models of continuous and discrete coupled oscillators --- the Kuramoto model (KM) \cite{acebron2005kuramoto}, Firefly Cellular Automata (FCA) \cite{lyu2015synchronization}, and Greenberg-Hastings model (GHM) \cite{greenberg1978spatial}.}

\commHL{Utilizing a few basic statistics of the underlying graphs, our method can achieve perfect accuracy when there is a significant difference in the topology of the underlying graphs between the synchronizing and the non-synchronizing examples (see Figures \ref{fig:toy_graph_dynamics} and \ref{fig:toy}). We find that when these graph statistics cannot separate the two classes of synchronizing and non-synchronizing very well (e.g., when the graphs are generated from the same random graph model, see Tables \ref{table:datasets} and \ref{table:datasets_big}), pairing a few iterations of phase configurations in the initial dynamics along with the graph statistics as the input to the classification algorithms can lead to significant improvement in accuracy. Our methods far surpass what is known by the half-circle concentration principle in classical oscillator theory (see our Methods section). We also find that in almost all such settings, dropping out the basic graph statistics and training our algorithms with only initial dynamics achieves nearly the same accuracy as with the graph statistics. Furthermore, we find that our methods are robust under using incomplete initial dynamics only observed on a few small subgraphs of large underlying graphs. }

\begin{figure*}[htpb]
	\centering

	\includegraphics[width=1\textwidth]{Figures/figure1.pdf}
	\caption{Sample points in the 30-node dynamics dataset for synchronization prediction. The heat maps show phase dynamics on graphs beneath them, where colors represent phases and time is measured by iterations from bottom to top (e.g. $t=0$ to $t=25$). Each example is labeled as `synchronizing' if it synchronizes at iteration 1758 for the Kuramoto model (70 for FCA and GHM)
	and `non-synchronizing' otherwise.
	Synchronizing examples have mostly uniform colors in the top row.  For training, only a portion of dynamics is used so that the algorithms rarely see a fully synchronized example (see Figure \ref{fig:dataset_visualization_training}).
	}
	\label{fig:dataset_visualization_full}
\end{figure*}

\vspace{-0.09cm}
\subsection*{Problem statement}

A graph $G=(V,E)$ consists of sets $V$ of nodes and $E$ of edges. Let $\Omega$ denote the \textit{phase space} of each node, which may be taken to be the circle $\mathbb{R}/2\pi \mathbb{Z}$ for continuous-state oscillators or the color wheel $\mathbb{Z}/\kappa \mathbb{Z}$, $\kappa\in \mathbb{N}$ for discrete-state oscillators. We call a map $X:V\rightarrow \Omega$ a \textit{phase configuration}, and say it is \textit{synchronized} if it takes a constant value across nodes (i.e., $X(v)=Const.$ for all $v\in V$). A \textit{coupling} is a function $\mathcal{F}$ that maps each pair $(G,X_{0})$ of graph and initial configuration $X_{0}:V\rightarrow \Omega$ deterministically to a \textit{trajectory} $(X_{t})_{t\ge 0}$ of phase configurations $X_{t}:V\rightarrow \Omega$. For instance, $\mathcal{F}$ could be the time evolution rule for the KM, FCA, or GHM.  Throughout the paper, $\mathbf{1}(\cdot)$ denotes the indicator function.  The main problem we investigate in this work is stated below:
\begin{description}
\item[$\bullet$] (Synchronization Prediction)
    \textit{Fix parameters $n\in \mathbb{N}$, $T\gg r>0$,  and coupling $\mathcal{F}$. Predict the following indicator function $\mathbf{1}(\text{$X_{T}$ is synchronized})$ given the initial trajectory $(X_{t})_{0\le t\le r}$ and optionally also with \commHL{statistics of} graph $G$.}
\end{description}

We remark that as $T$ tends to infinity, the indicator in the problem statement converges to the following indicator function $\mathbf{1}(\text{$X_{t}$ is eventually synchronized})$ which aligns more directly with the initial questions \ref{Q1} and \ref{Q2} than the indicator in the above problem statement, but determining whether a given system will never synchronize in amounts to finding a non-synchronizing periodic orbit, which is computationally infeasible in general.
See Figure \ref{fig:dataset_visualization_training} for an illustration of the synchronization prediction problem.

\subsection*{Related works}
\label{subsection:related_works}

 There are a number of recent works incorporating machine learning methods to investigate problems on coupled oscillators or other related dynamical systems, which we briefly survey in this section and summarize in Table \ref{table:comparison}.

 \commHL{In Fan et al.\cite{fan2021anticipating}, the authors are concerned with identifying the critical coupling strength at which a system of coupled oscillators undergo a phase transition into synchronization, where the underlying graph consists of 2-4 nodes with a fixed topology (e.g., a triangle or a star with three leaves). Guth et al. \cite{guth2019machine} use binary classification methods with surrogate optimization (SO) in order to learn optimal parameters and predictors of extreme events. Their work is primarily concerned with learning whether or not intermittent extreme events will occur in various 1D or 2D partial differential equation models.
 Similarly, Chowdhury et al. \cite{chowdhury2021extreme} utilize a long-short term memory (LSTM) \cite{hochreiter1997long} network to predict whether or not an extreme event will occur on globally coupled mean-field logistic maps on complete graphs. Thiem et al.\cite{Thiem2020emergent} use Feed-forward neural networks (FFNN) \cite{bishop2006pattern} to learn coarse-grained dynamics of Kuramoto oscillators and recover the classical order parameter. Biccari et al.\cite{Biccari2020stochastic} use gradient descent (GD) and the random batch method (RBM) to learn control parameters to enhance the synchronization of Kuramoto oscillators. Slightly less related work is Hefny et al.\cite{hefny2015supervised}, where the authors use hidden Markov models, LASSO regression, and spectral algorithms for learning lower-dimensional state representations of dynamical systems and apply their method to a knowledge tracing model for a dataset of students' responses to a survey. }

\begin{table}[htbp]
		\centering

		\begin{tabular}{c|ccccccc}
			\textit{References} & $\#$ nodes & $\#$ graphs  &\# configs. & model & ML &  Goal \\
			\hline
			\rule{0pt}{1.1\normalbaselineskip} Fan et al. \cite{fan2021anticipating} & 2-4 & 1 & 1 & Lorenz, KM & FFNN  & Phase transition \\
			\rule{0pt}{1.1\normalbaselineskip} Guth et al. \cite{guth2019machine} & N/A & N/A & 1 & 1D and 2D PDE &  SO  & Extreme events \\
			\rule{0pt}{1.1\normalbaselineskip} Chowdhury et al.\cite{chowdhury2021extreme} & 200 & 1 & 1 & $\begin{matrix} \textup{Mean-field} \\ \textup{logistic map}  \end{matrix}$ & LSTM & Extreme events \\
		    \rule{0pt}{1.1\normalbaselineskip} Thiem et al.\cite{Thiem2020emergent} & 1500-8000 & 1 & 2000 & KM & FFNN & Parameter estimation\\
		    \rule{0pt}{1.1\normalbaselineskip} Biccari et al.\cite{Biccari2020stochastic} & 10-1000 & 1 & 1 & KM & GD, RBM & Parameter estimation\\
		    \rule{0pt}{1.1\normalbaselineskip}  Itabashi et al.\cite{itabashi2021evaluating} & 128-256 & 1  &100 & KM & TDA & $\begin{matrix} \textup{Synchronization}/ \\ \textup{Non-synchronization}/ \\ \textup{Chimera state} \end{matrix} $ \\
		    \hline
		    &&&& KM, & RF, GB, &  \text{Synchronization}/ \\
		    This work & 15-600 & \textbf{2K-200K} & 1 & FCA, & FFNN, & \textup{Non-synchronization}\\
		    &&&& GHM & LRCN \\
		\end{tabular}%

		\caption{Comparison of settings in related works on learning coupled oscillator dynamics using machine learning methods. Recent works in the table focus on learning features of dynamics on fixed graphs. \commHL{In contrast, we aim to classify the long-term dynamics of a given system on a diverse set of underlying graphs.} The column for `\# configs.' refers to the number of distinct initial phase configurations considered for each graph in training
		}
		\label{table:comparison}
	\end{table}

\commHL{ On the other hand, Itabashi et al.\cite{itabashi2021evaluating} consider classifying coupled Kuramoto oscillators according to their future dynamics from certain features derived (using topological data analysis (TDA)) from their early-stage dynamics (or `initial dynamics' in our terminology). As in the other references above, the underlying graph is fixed in each classification task (a complete graph with weighted edges, which may have two or four communities). But unlike in the references above, the initial configuration, instead of the model parameter, is varied to generate different examples on the same underlying graph. The authors observed that some long-term dynamical properties (e.g., multi-cluster synchronization) can be predicted by the derived features of the initial dynamics. This point is consistent with one of our findings that the first few iterations of the initial dynamics may contain crucial information in predicting the long-term behavior of coupled oscillator systems. }

 \commHL{While sharing high-level ideas and approaches with the aforementioned works, our work has multiple distinguishing characteristics in the problem setting and approaches. First, in all of the aforementioned works, a dynamical system on a fixed underlying graph is considered. But in our setting, there are as many as 200K non-isomorphic underlying graphs and we seek to predict whether a given system of oscillators on highly varied or even unknown graphs will eventually synchronize or not. \commHL{Furthermore, we also consider the case when machine learning algorithms are trained on partial observation (e.g., initial dynamics restricted on some subgraphs). }
 Second, only in our work, are discrete models of coupled oscillators (e.g., FCA and GHM) considered, whereas only models with continuous phase space (e.g., the KM) are considered in the literature. Third, only in our work, is the classical concentration principle (a.k.a. the `half-circle concentration', see our Methods section) in the oscillator theory used as a rigorous benchmark to evaluate the efficacy of employed machine learning methods. Finally, we employ various binary classification algorithms such as Random Forest (RF) \cite{breiman2001random} Gradient Boosting (GB)\cite{friedman2002stochastic},  Feed-forward Neural Networks (FFNN) \cite{bishop2006pattern}, and our own adaptation of a Long-term Recurrent Convolutional Networks (LRCN) \cite{donahue2015long} . }

 We remark that there are a number of cases where rigorous results are available for the question of predicting the long-term behavior of coupled oscillators on a graph $G$ and initial configuration $X_{0}$. For instance, the $\kappa=3$ instances of GHM and another related model called Cyclic Cellular Automata (CCA) \cite{fisch1990cyclic} have been completely solved \cite{gravner2018limiting}. Namely, given the pair $(G,X_{0})$, the trajectory $X_{t}$ synchronizes eventually if and only if the discrete vector field on the edges of $G$ induced from $X_{0}$ is conservative (see \cite{gravner2018limiting} for details). Additionally, the behavior of FCA on finite trees is also well-known: given a finite tree $T$ and $\kappa\in \{3,4,5,6\}$, every $\kappa$-color initial configuration on $T$ synchronizes eventually under $\kappa$-color FCA if and only if the maximum degree of $T$ is less than $\kappa$; for $\kappa\ge 7$, this phenomenon does not always hold\cite{lyu2015synchronization, lyu2016phase}. \commHL{This theoretical result on FCA was used in the experiment in Figures \ref{fig:toy_graph_dynamics} and \ref{fig:toy}}. Furthermore, there is a number of works on the clustering behavior of these models on the infinite one-dimensional lattice, $\mathbb{Z}$ (FCA\cite{lyu2015synchronization,lyu2019persistence}, CCA\cite{fisch1990one, fisch1991cyclic, fisch1992clustering,lyu2019persistence} and GHM \cite{lyu2019persistence, durrett1991some}).

\section*{Methods}

The pipeline of our approach is as follows. Namely, 1) fix a model for coupled oscillators; 2) generate a \textit{dynamics dataset} of a large number of non-isomorphic graphs with an even split of synchronizing and non-synchronizing dynamics; 3) train a selected binary classification algorithm on the dynamics dataset to classify each example (initial dynamics with or without underlying features of the graph) into one of two classes, `synchronizing' or `non-synchronizing'; 4) validate the accuracy of the trained algorithms on fresh examples by comparing the predicted behavior of the true long-term dynamics. We use the following classification algorithms: Random Forest (RF) \cite{breiman2001random} Gradient Boosting (GB)\cite{friedman2002stochastic},  Feed-forward Neural Networks (FFNN) \cite{bishop2006pattern}, and our own adaptation of Long-term Recurrent Convolutional Networks (LRCN) \cite{donahue2015long} which we call the \textit{GraphLRCN} (further information such as implementation details and hyperparameters can be found in the SI).

As a baseline for our approach, we use a variant of the well-known ``concentration principle'' in the literature on coupled oscillators. Namely, regardless of the details of graph structure and model, synchronization is guaranteed if the phases of the oscillators are concentrated in a small arc of the phase space \textit{at any given time} (see the next subsection). This principle is applied at each configuration up to the training iteration used to train the binary classifiers.

For question \ref{Q3} on synchronization prediction on the initial dynamics partially observed on subgraphs, as well as reducing the computational cost of our methods for answering \ref{Q1} and \ref{Q2}, we propose an ``ensemble prediction'' algorithm (Algorithm \ref{algorithm:collective_predictor}) that scales up our method to large graphs by training on dynamics observed from multiple random subgraphs. Namely, suppose we are to predict the dynamics of some connected $N$-node graphs, where only the initial dynamics are observed on a few small connected subgraphs of $n \ll N$ nodes. We first train a binary classification algorithm on the dynamics observed from those subgraphs and then aggregate the predictions from each subgraph (e.g., using a majority vote) to get a prediction for the full dynamics on $N$ nodes.

\subsection*{The concentration principle for synchronization and baseline predictor}
In the literature on coupled oscillators, there is a fundamental observation that concentration (e.g., into an open half-circle) of the initial phase of the oscillators leads to synchronization for a wide variety of models on arbitrary connected graphs (see, e.g., \cite[Lem 5.5]{lyu2018global}). This is stated in the following bullet point for the KM and FCA and we call it the ``concentration principle''. This principle has been used pervasively in the literature of clock synchronization \cite{nishimura2011robust, klinglmayr2012guaranteeing, proskurnikov2016synchronization, nunez2014synchronization}  and also in multi-agent consensus problems \cite{moreau2005stability, papachristodoulou2010effects, chazelle2011total}.
\begin{description}
    \item[$\bullet$] (Concentration Principle) \textit{Let $G$ be an arbitrary connected graph. For the Kuramoto model (see eq. (1) in SI) with identical intrinsic frequency and for FCA (see eq. (3) in SI), given dynamics on $G$ synchronize if all phases at any given time are confined in an open half-circle in the phase space $\Omega$. Furthermore, if all states used in the time-$t$ configuration $X_{t}$ are confined in an open half-circle for any $t\ge 1$, then the trajectory on $G$ eventually synchronizes.}
\end{description}
The `open half-circle' refers to any arc of length $<\pi$ for the continuous \commHL{phase} space $\Omega=\mathbb{R}/2\pi \mathbb{Z}$ and any interval of $<\kappa/2$ consecutive integers ($\textup{mod}\,\, \kappa$) for the discrete \commHL{phase} space $\Omega=\mathbb{Z}/\kappa \mathbb{Z}$. This is a standard fact known to the literature and it follows from the fact that the couplings in the statement monotonically contract given any initial phase configuration under the half-circle condition toward synchronization. It is not hard to see the above half-circle concentration does not hold for GHM. Accordingly, for GHM, we say a phase configuration $X_{t}$ is \textit{concentrated} if $X_{t}$ is synchronized.

We now introduce the following baseline synchronization predictor: Given $(X_{t})_{0\le t \le r}$ and $T>r$,
\begin{description}
    \item[$\bullet$] (Baseline predictor) \textit{Predict synchronization of $X_{T}$ if $X_{t}$ is concentrated for any $1\le t \le r$. Otherwise, flip a fair coin. }
\end{description}
Notice that the baseline predictor never predicts synchronization incorrectly if $X_{r}$ is concentrated. For non-concentrated cases, the baseline does not assume any knowledge and gives a completely uninformed prediction. Quantitatively, suppose we have a dataset where $\alpha$ proportion of \commHL{entire} samples are synchronizing \commHL{(in all our datastes, $\alpha=0.5$)}. Suppose we apply the baseline predictor where we use the first $r$ iterations of dynamics for each sample. Let $x=x(r)$ denote the proportion of synchronizing samples \commHL{that concentrate by iteration $r$ among all among all synchronizing samples.} Then the baseline predictor's accuracy is given by $x\alpha + (1-\alpha+(1-x)\alpha)/2=0.5+x\alpha/2$, where the term $x\alpha/2$ can be regarded as the gain obtained by using the concentration principle layered onto the uninformed decision.

\iffalse
\begin{align}
&1-\alpha + x\alpha \\
&x\alpha + (1-\alpha+(1-x)\alpha)/2=(1+x\alpha)/2 \\
L-P &= 0.5 -\alpha +  x \left( \frac{3\alpha-1}{2}  \right)
\end{align}
\fi

\subsection*{An illustrative example: the Kuramoto model on complete vs. ring; GHM on path vs. complete; FCA on tree vs. ring}

We first give a simple example to illustrate our machine learning approach to the synchronization prediction problem. More specifically, we create \commHL{datasets} of an equal number of synchronizing and non-synchronizing examples, where there is a significant difference in the topology of the underlying graphs between the synchronizing and the non-synchronizing examples. In such a setting, one can expect that knowing the basic graph features --- the number of edges, min/max degree, diameter, and the number of nodes --- will be enough to distinguish between synchronizing and non-synchronizing examples.

\begin{figure*}[htpb]
	\centering

	\includegraphics[width=1\textwidth]{Figures/figure2.png}
	\caption{ Examples of synchronizing and non-synchronizing dynamics of Kuramoto, Greenberg-Hastings, and FCA oscillators on 30-node graphs with two topologies (complete vs. ring, path vs. complete, and tree vs. ring, respectively). The heat maps show phase dynamics on graphs beneath them, where colors represent phases and time is measured by iterations from bottom-to-top (e.g. $t=0$ to $t=70$). Synchronizing examples have uniform color in the top row. The horizontal bar indicates the `training iteration', which is the maximum number of iterations in the initial dynamics fed into the classification algorithm for prediction.
	}
	\label{fig:toy_graph_dynamics}
\end{figure*}

\commHL{For each of the coupled oscillator models of the KM, FCA, and GHM, we create a dataset that consists of 1K synchronizing examples and 1K non-synchronizing examples on 30 node graphs, where all 1K examples in each of the two classes share the same underlying graph topology. For the KM, the synchronizing and the non-synchronizing examples are on a complete graph and on a ring, respectively. For GHM (with $\kappa=5$), the synchronizing and the non-synchronizing examples are on a path and on a complete graph, respectively. Lastly for FCA (with $\kappa=5$), the synchronizing and the non-synchronizing examples are on a tree with a maximum degree at most four and on a ring, respectively. Our choice of these graph topologies reflects rigorously established results in the literature. Namely, it is well-known that coupled Kuramoto oscillators (with identical intrinsic frequencies) on a complete graph will always synchronize, and one can easily generate a non-synchronizing initial configuration for Kuramoto oscillators on a ring \cite{strogatz2000kuramoto,acebron2005kuramoto}. For GHM, in the SI, we prove that a $\kappa$-color (arbitrary $\kappa \geq 3$) GHM on a path of length $n$ will always synchronize by iteration $n+\kappa$, regardless of the initial configuration. Lastly for FCA, it is known that FCA with $\kappa=5$ color dynamics always synchronize on trees with maximum degree at most four \cite{lyu2015synchronization}. The results of our experiments for these three datasets are shown in Figure \ref{fig:toy}.}

\commHL{For the three datasets described above, we achieve perfect classification accuracy with FFNN of distinguishing synchronizing vs non-synchronizing by using the following five basic statistics of the underlying graphs ---
the number of edges, min/max degree, diameter, and the number of nodes  --- as expected by the theoretical results we mentioned above. Additionally, we hide the graph statistics completely and train FFNN only using the initial dynamics up to a variable training iteration. As we feed in more initial dynamics for training, FFNN quickly improves in prediction accuracy, far more rapidly than the baseline does. This indicates that FFNN may be quickly learning some latent distinguishing features from the initial dynamics that are more effective than the half-circle concentration employed by the baseline predictor.}

\begin{figure}[H]
	\centering

	\includegraphics[width=1\textwidth]{Figures/dynvfeat.png}
	\caption{\commHL{Binary classification accuracies of synchronizing vs. non-synchronizing dynamics when the underlying graphs in either of these two classes share the same topology. For the Kuramoto model, we compare complete graphs (synchronizing) to rings (non-synchronizing), for the GHM model we compare rings (synchronizing) vs complete graphs (non-synchronizing), and for FCA we compare trees with maximum degree 4 (synchronizing) to rings (non-synchronizing). FFNN (Features) uses FFNN as the binary classification algorithm with five basic graph features --- the number of edges, min/max degree, diameter, and the number of nodes --- as the input, which achieves perfect classification in all cases. FFNN (Dynamics) uses FFNN as the binary classification algorithm with a varying number of iterations of initial dynamics as input (specified as the horizontal axis), without any information about the underlying graph. The baseline predictor uses the concentration principle (see the main text for more details). Notice that FFNN (Dynamics) far surpasses the baseline.}

	}
	\label{fig:toy}
\end{figure}

\subsection*{Generating the dynamics datasets}

\commHL{In the example we considered in Figures \ref{fig:toy_graph_dynamics} and \ref{fig:toy}, there was a clear distinction between the topology of the underlying graphs that were synchronizing and the graphs that were non-synchronizing, and basic graph statistics can yield perfect classification accuracy through FFNN as the choice of the binary classifier. In this section, we consider datasets for which it is much more difficult to classify based only on the same graph statistics. The way we do so is by generating a large number of underlying graphs from the \textit{same} random graph model with the \textit{same} parameters. In this way, even though individual graphs realized from the random graph model may have different topologies, graph statistics such as edge density or maximum degree are concentrated around their expected values. This is in contrast to the set of underlying graphs in the three datasets we considered in Figures \ref{fig:toy_graph_dynamics} and \ref{fig:toy}, where half of them are isomorphic to each other and another half are also isomorphic to each other. For the datasets we generate in this way, which we will describe in more detail shortly after, classifying only with the basic graph statistics achieves an accuracy of 60-70\% (in comparison to 100\% as before).
}

\commHL{We generate a total of tweleve datasets described in Tables \ref{table:datasets} and \ref{table:datasets_big} as follows.} Data points in each dataset consist of three statistics computed for a pair $(G,X_{0})$ of an underlying graph, $G=(V,E)$, and initial configuration, $X_{0}:V\rightarrow \Omega$: 1) first $r$ iterations of dynamics $(X_{t})_{0\le t \le r}$ (using either the KM, FCA, or GHM), (optional) 2) features of $G$ and $X_{0}$, and 3) the label that indicates whether $X_{T}$ is concentrated or not. We optionally include the following six features: \textit{number of edges}, \textit{min degree}, \textit{max degree}, \textit{diameter}, \textit{number of nodes}, and \textit{quartiles of initial phases in $X_{0}$}. We say a data point is `synchronizing' if the label is 1, and `non-synchronizing,' label 0, otherwise. Every dataset we generate contains an equal number of synchronizing and non-synchronizing examples, and the underlying graphs are all connected and non-isomorphic \commHL{(as opposed to the datasets in Figures \ref{fig:toy_graph_dynamics} and \ref{fig:toy})}.

    To generate a single $n$-node graph, we use an instance of the Newman-Watts-Strogatz (NWS) model\cite{newman2002random}, which originally has three
    parameters $n$ (number of nodes), $p$ (shortcut edge probability) and $k$ (initial degree of nodes; We use the implementation in \texttt{networkx} python package, see \cite{hagberg2008exploring}), with an added integer parameter $M$ (number of calls for adding shortcut edges). Namely, we start from a cycle of $n$ nodes, where each node is connected to its $k$ nearest neighbors. Then we attempt to add a new edge between each initial non-edge $(u,v)$ with probability $p/(n-k-3)$ independently $M$ times. The number of new edges added in this process follows the binomial distribution with $\binom{n}{2} - \frac{nk}{2}$ trials with success probability $\left(1 - \left(1-\frac{p}{n-k-1} \right) \right)^{M}\approx pm/(n-k-1)$. This easily yields that the expected number of edges in our random graph model is  $\frac{nk}{2}  +  \frac{n^{2}pM}{2(n-k-1)} + O(k)$.

\begin{table}[htbp]
	\centering

	\begin{tabular}{ccccccc}
		\hline
		 \textit{Datasets} & $\dataset{KM}_{15}$ & $\dataset{KM}_{30}$ & $\dataset{FCA}_{15}$ & $\dataset{FCA}_{30}$ & $\dataset{GHM}_{15}$ & $\dataset{GHM}_{30}$  \\
		\hline
		\# nodes & 15 & 30 & 15 & 30 &15 & 30 \\
		avg of \# edges & 29.65 &57.49 & 23.91 & 47.45 & 22.88& 48.18 \\
		std of \# edges & 3.42 & 5.67 & 2.34& 4.15 & 2.35& 4.11 \\
		avg diameter & 4.32 & 7.00 & 4.32 & 6.01 & 4.30 & 6.12 \\
		std of diameter & 0.68 & 1.29 & 0.67 & 0.95 & 0.66 & 0.98\\
        $r$ (training iter) & 126 & 126 & 25 & 25 & 25 & 25 \\
        $T$ (prediction iter) & 1758 &  1758 & 70 & 70 & 70 & 70 \\
        \# Sync.   & 100K & 40K & 100K & 40K & 100K & 40K\\
        \# Nonsync.  & 100K & 40K & 100K & 40K & 100K & 40K\\

        \hline
	\end{tabular}%
	\caption  {
Dynamics datasets were generated for three models with two node counts. In each dataset, all graphs are connected and non-isomorphic. $\#$ Sync. denotes the number of examples in the dataset such that the phase configuration $X_{T}$ at iteration $T$ is concentrated}
	\label{table:datasets}
\end{table}

\begin{figure*}[htpb]
	\centering

	\includegraphics[width=1\textwidth]{Figures/figure4.pdf}
	\caption{Sample points in the 30-node training data set for synchronization prediction. The full dataset consists of 40K synchronizing and 40K non-synchronizing 30-node connected non-isomorphic graphs and dynamics on them for each of the three models the KM, FCA, and GHM (\commHL{see} Table \ref{table:datasets}).
	}
	\label{fig:dataset_visualization_training}
\end{figure*}

    \iffalse
    \begin{align}
        \mathbb{E}[G(n,k,p,M)]&=\frac{nk}{2} + \mathbb{E}\left[ \sum_{u\nsim v}  \mathbf{1}(\textup{$uv$ filled in after $M$ calls})\right] \\
        &= \frac{nk}{2}  + \left(\binom{n}{2} - \frac{nk}{2}\right) \left(1 - \left(1-\frac{p}{n-k-1} \right)^{M} \right) \\
        &= \frac{nk}{2}  +  \left(\binom{n}{2} - \frac{nk}{2}\right) \frac{pM}{n-k-1} + O(1) \\
        &= \frac{nk}{2}  +  \frac{n^{2}}{2} \frac{pM}{n-k-1} + O(k) \\
    \end{align}
    \fi

    The NWS model of random graphs is known to exhibit the `small world property', which is to have relatively small mean path lengths in contrast to having a low local clustering coefficient. This is a widely observed property of many real-world complex networks \cite{newman2002random} --- as opposed to the commonly studied Erd\"os-R\'eyni graphs. It is known that many models of coupled oscillators have a hard time synchronizing when the underlying graph is a ring, as the discrepancy between oscillators tends to form traveling waves circulating on the ring \cite{lee2008experiments}. On the other hand, it is observed that coupled oscillator systems on dense graphs are relatively easier to synchronize. For instance, Kassabov, Strogatz, and Townsend recently showed that Kuramoto oscillators with an identical natural frequency on a connected graph where each node is connected to at least 3/4 of all nodes are globally synchronizing for almost all initial configurations \cite{kassabov2021sufficiently}. Since we intend to generate both synchronizing and non-synchronizing examples to form a balanced dataset, it is natural for us to use a random graph model that sits somewhere between rings and dense graphs. In this sense, NWS is a natural choice for a random graph model for our purpose. Using other models such as Erd\"os-R\'eyni, for example, \commHL{for} generating the balanced dataset of synchronizing and non-synchronizing examples as in Tables \ref{table:datasets} and \ref{table:datasets_big}, \commHL{is} computationally very demanding.

    In Table \ref{table:datasets}, we make note of the average graph edge counts and standard deviations of the random graphs that were used to simulate our models. These characteristics, average edge count and standard deviation of edges counts, correspond to the clustering of edges in the graph, which intuitively affects the propagation of information or states in our cellular automata.

Also in Table \ref{table:datasets}, we give a summary of the six datasets on the three models for two node counts $n=15,30$, each with 200K and 80K examples, respectively, which we refer to as $\dataset{KM}_{n}$, $\dataset{FCA}_{n}$, $\dataset{GHM}_{n}$ for $n=15,30$. Underlying graphs are sampled from the NWS model with parameters $n\in \{15,30\}$, $M=1$, and $p=0.85$ for the KM and $p=0.65$ for FCA and GHM. In all cases, we generated about 400K examples and subsampled 200K and 80K examples for $n=15,30$, respectively, so that there are an equal number of synchronizing and non-synchronizing examples, with all underlying graphs as non-isomorphic. The limits for both sets were chosen by memory constraints imposed by the algorithms used. To give a glance at the datasets, we provide visual representations. In Figure \ref{fig:dataset_visualization_training}, we show five synchronizing and non-synchronizing examples in $\dataset{KM}_{30}$, $\dataset{FCA}_{30}$, and $\dataset{GHM}_{30}$. \\

\begin{table}[htbp]
	\centering
	\begin{tabular}{ccccccc}
		\hline
		 \textit{Datasets} & $\dataset{FCA}_{600} $& $\dataset{FCA}_{600}'$ & $\dataset{FCA}_{600} ''$ & $\dataset{KM}_{600}$ & $\dataset{KM}_{600}'$ & $\dataset{KM}_{600} ''$\\
		\hline
		\# nodes & 600 & 600 & 300-600 &  600 & 600 & 300-600\\
		std of \# nodes & 0 & 0 & 86.60 &  0 & 0 & 86.60 \\
		avg of \# edges &2985.53&4749.24&2799.49 &1051.08&1109.85&757.89\\
		std of \# edges &37.85&2371.72& 1461.08&19.66&79.47&56.81\\
        $r$  & 50 & 50 & 50& 400 & 400 & 400 \\
        $T$ & 600 &  600 & 600 & 1758 & 1758 & 1758\\
        \# Sync.   & 1K & 1K & 1K & 1K & 1K & 1K \\
        \# Nonsync.  & 1K & 1K & 1K & 1K & 1K & 1K \\

        \hline
	\end{tabular}%
	\caption{Dynamics datasets generated for FCA and Kuramoto on 600 nodes ($\dataset{FCA}_{600}$, $\dataset{KM}_{600}$, $\dataset{FCA}_{600}'$, $\dataset{KM}_{600}'$) and on 300-600 nodes ($\dataset{FCA}_{600}''$, $\dataset{KM}_{600}''$). In each dataset, all graphs are connected and non-isomorphic. $\#$ Sync. denotes the number of examples in the dataset such that the phase configuration $X_{T}$ at iteration $T$ is concentrated. Here $r$ and $T$ refers to the training and the prediction iterations defined in Table \ref{table:datasets}.}
	\label{table:datasets_big}
\end{table}

We also generated six dynamics datasets with a larger number of nodes on FCA and Kuramoto dynamics, as described in Table \ref{table:datasets_big}. The fixed node datasets $\dataset{FCA}_{600}$, $\dataset{FCA}_{600}'$, $\dataset{KM}_{600}$, and $\dataset{KM}_{600}'$ each consist of 1K synchronizing and non-synchronizing examples of FCA and Kuramoto dynamics on non-isomorphic graphs of 600 nodes. The underlying graphs for the fixed node FCA datasets are generated by the NWS model with parameters $n=600$, $p=0.6$ and $N=5$ for $\dataset{FCA}_{600}$ and $p\sim\textup{Normal(}\mu_X,0.04)$, $\mu_X\sim\textup{Uniform}(0.32,0.62)$, for each $N\sim \textup{Uniform}(\{1,2,\dots,20\})$ calls for $\dataset{FCA}_{600}'$. Similarly, to generate the fixed node Kuramoto datasets, $\dataset{KM}_{600}$ and $\dataset{KM}_{600}'$, we used the NWS model with parameters $n=600$, $p=0.15$ and $N=5$ for $\dataset{KM}_{600}$ and $p\sim\textup{Normal(}\mu_X,0.04)$, $\mu_X\sim\textup{Uniform}(0.32,0.62)$, for each $N\sim \textup{Uniform}(\{1,2,\dots,20\})$ calls for $\dataset{KM}_{600}'$. Consequently, the number of edges in the graphs from $\dataset{KM}_{600}$ and $\dataset{FCA}_{600}$ are sharply concentrated around its mean whereas $\dataset{KM}_{600}'$ and $\dataset{FCA}_{600}'$ have much greater overall variance in the number of edges (see Table \ref{table:datasets_big}). For the varied node datasets, $\dataset{FCA}_{600}''$ and $\dataset{KM}_{600}''$, we kept $p$ distributed in the same way as done for $\dataset{KM}_{600}'$ $\dataset{FCA}_{600}'$, but for each $N\sim \textup{Uniform}(\{1,2,\dots,20\})$ calls of adding shortcut edges, but additionally varied the number of nodes as $n\sim \textup{Uniform}(\{300,301,\dots,600\})$. In this case, both the number of nodes and edges have relatively greater variation compared to the other datasets.

We omit the GHM from this experiment because the dynamics are extremely prone to non-synchronization as a network has more cycles\cite{durrett1993asymptotic}. Hence, for these large graphs, almost all \commHL{GHM dynamics} will be non-synchronizing. \commHL{For instance, for the same set of networks we used to generate the $\dataset{FCA}_{600}$ dataset (consisting of 40K networks of 600 nodes), none of the GHM dynamics synchronized. Hence, there is no meaningful} classification problem to be discussed since the presence of synchronizing graphs that follow this model is extremely sparse.

\subsection*{Scaling up by learning from \commHL{subgraphs}}

In this subsection, we discuss a way to extend our method for the dynamics prediction problem simultaneously in two directions; 1) larger graphs and 2) a variable number of nodes. The idea is to train our dynamics predictor on subsampled dynamics of large graphs (specified as induced subgraphs and induced dynamics), and to combine the local classifiers to make a global prediction.  In the algorithm below, $f(X_{T}):=\mathbf{1}(\textup{$X_{T}$ is concentrated})$, and if $X_{t}$ is a phase configuration on $G=(V,E)$ and $G_{i}=(V_{i},E_{i})$ is a subgraph of $G$, then $X_{t}|_{G_{i}}$ denotes the restriction $v\mapsto X_{t}(v)$ for all $v\in V_{i}$.

\begin{algorithm}[H]
	\caption{Ensemble Prediction of Synchronization}
	\label{algorithm:collective_predictor}
	\begin{algorithmic}[1]
		\State \textbf{Input:} Dynamics dataset on graphs with $\ge N$ nodes; \,\, Test point $(G', (X'_{t})_{0\le t \le r})$;
		\State \textbf{Parameters:} $n_{0}\le N$ (size of subgraphs),\,\, $k_{\textup{train}}$, $k_{\textup{test}}$ ($\#$ of subgraphs) ,\,\, $\theta$ (prediction threshold)
		\State \textbf{Subsample Dynamics:}
		\State \quad For each data point  $(G,(X_{t})_{0\le t \le r}, f(X_{T}))$:
		\State \quad\quad Sample $n_{0}$-node connected subgraphs $G_{1},\dots,G_{k_{\textup{train}}}$ of $G$;
		\State \qquad Form restricted triples $(G_{i}, (X_{t}|_{G_{i}})_{0\le t\le r}, f(X_{T}))$
		\State \textbf{Train Dynamics Predictor:}
		\State \quad Train a binary classifier on the restricted triples;
		\State \textbf{Ensemble Prediction:}
		\State \quad Sample $n_{0}$-node connected subgraphs $G_{1}',\dots,G_{k_{\textup{test}}}'$ of $G'$;
		\State \quad $\hat{f}:=$ mean of predictions of $f(X_{T})$ on subdynamics on $G_{i}'$'s
		\State \textbf{Output:} $\mathbf{1}(\hat{f}>\theta)$
	\end{algorithmic}
\end{algorithm}
\section*{Results}

Regardless of the three models of coupled oscillators and selected binary classification algorithm, we find that our method used to address \ref{Q1} and \ref{Q2} on average shows at least a 30\% improvement in prediction performance compared to this concentration-prediction baseline for dynamics on 30 node graphs. In other words, our results indicate that the concentration principle applied at each configuration is too conservative in predicting synchronization, and there might be a generalized concentration principle that uses the whole initial dynamics, which our machine learning methods seem to learn implicitly.

Using Algorithm \ref{algorithm:collective_predictor} for \ref{Q3}, we achieve an accuracy score of over 85\% for predicting the commonly studied Kuramoto model on 600 node graphs by only using four 30-node subgraphs where the corresponding baseline gets $55\%$ accuracy. In particular, we observe the baseline with locally observed initial dynamics tends to misclassify non-synchronizing examples as synchronizing, as locally observed dynamics can concentrate in a short time scale while the global dynamics do not.

\subsection*{Synchronization prediction accuracy for 15-30 node graphs}

We apply the four binary classification algorithms for the six 15-30 node datasets described in Table \ref{table:datasets} in order to learn to predict synchronization. Each experiment uses initial dynamics up to a variable number of training iterations $r$ that is significantly less than the test iteration $T$, and the goal is to predict whether each example in the dataset is synchronized at an unseen time $T$. We also experiment with and without the additional graph information described in Table \ref{table:datasets} in order to investigate the main questions \ref{Q1} and \ref{Q2}, respectively. We plot prediction accuracy using four classification algorithms (RF, GB, FFNN, LRCN) and the baseline predictor versus the training iteration $r$, with and without the graph features. The problem of synchronization prediction becomes easier as we increase the training iteration $r$, as indicated by the baseline prediction accuracy in Figure \ref{fig:full_plot_accuracy}. For instance, for $\dataset{KM}_{30}$, there is no trivially synchronizing example at iteration $0$, and but about 10$\%$ of the synchronizing examples will have become phase-concentrated by iteration $r=25$.

Now we discuss the results in Figure \ref{fig:full_plot_accuracy}. \commHL{As we intended before, for the six datasets in Table \ref{table:datasets}, classifying only with the basic graph statistics achieves an accuracy of 60-70\% in comparison to 100\% as in the experiments in Figures \ref{fig:toy_graph_dynamics} and \ref{fig:toy}. This is shown by the gray horizontal lines in Figure \ref{fig:full_plot_accuracy}, which indicates the classification accuracy of FFNN trained only with the same five basic graph statistics used before. This indicates that we may need to use more information on the data points in order to obtain improved classification accuracy. One way to proceed is to use more graph statistics such as clustering coefficients\cite{watts1998collective}, modularity\cite{newman2013spectral}, assortativity \cite{allen2017two}, eigenvalues of the graph Laplacian \cite{zhang2011laplacian}, etc. }

\commHL{Instead, aiming at investigating Question \ref{Q1}, we proceed by additionally using dynamics information, meaning that we include the initial dynamics, $(X_{t})_{0\le t \le r}$, up to a varying number of training iteration $r$.  These results are shown in the first and the third columns in Figure \ref{fig:full_plot_accuracy}. At $r=0$, the input consists of five graph statistics we considered before --- \textit{number of edges}, \textit{min degree}, \textit{max degree}, \textit{diameter}, \textit{number of nodes} --- as well the initial configuration $X_{0}$ with its quartiles. In all cases, all four binary classifiers trained with initial dynamics significantly outperform the \commHL{concentration principle baseline.} The classifiers RF, GB, and FFNN show similar performance in all cases. On the other hand, the GraphLRCN in some cases outperforms the other classifiers, especially so with GHM on 30 nodes. For instance, when $r=20$, $10$ and $4$ for $\dataset{KM}_{30}$, $\dataset{FCA}_{30}$ and $\dataset{GHM}_{30}$, respectively, GraphLRCN achieves a prediction accuracy of $73\%$ (baseline $55\%$, $1.25\%$ concentrates), $84\%$ (baseline $52\%$, $1\%$ concentrates) and $96\%$ (baseline $50\%$, $0\%$ concentrates), respectively.
}

\commHL{
Now that we have seen that initial dynamics can be used along with the basic graph features to gain a significant improvement in classification performance, we take a step further and see how accurate we can be by only using the initial dynamics, in relation to Question \ref{Q2}. That is, we drop all graph-related features from the input, and train the binary classifiers only with the initial dynamics up to varying iteration $r$. The results are reported in the second and the fourth columns in Figure \ref{fig:full_plot_accuracy}. Since now the classifiers are not given any kind of graph information at all, one might expect a significant drop in the performance. However, surprisingly, except for the dataset $\dataset{KM}_{30}$, the classification accuracies are almost unchanged after dropping graph features altogether from training.
}

\begin{figure*}[htpb]
    \centering
     \includegraphics[width=1\linewidth]{Figures/figure5_new.pdf}
     \vspace{-0.5cm}
    \caption{Synchronization prediction accuracies of four machine learning algorithms for the KM, FCA, and GHM coupled oscillators synchronization. For each of the six datasets in Table \ref{table:datasets}, we used 5-fold cross-validation with 80/20 split of test/train. Accuracy is measured by the ratio of the number of correctly classified examples to the total number of examples. Algorithms for the second and the fourth columns are trained only with dynamics up to various iterations $r$ indicated on the horizontal axes, whereas the other two columns also use additional graph features. \commHL{$r=0$ in the "with Features" columns indicates the input consisting of initial phase configuration $X_0$ and its \textit{quartiles} paired with the five graph statistics of \textit{number of edges}, \textit{min degree}, \textit{max degree}, \textit{diameter}, \textit{number of nodes}. The gray dashed line represents training FFNN only on the five graph statistics. Hence its accuracy is constant with respect to the varying number of training iterations. } }
    \label{fig:full_plot_accuracy}
\end{figure*}

\commHL{
In addition, note that for $\dataset{FCA}_{15}$ and $\dataset{FCA}_{30}$, training all four classifiers on only training iterations of $r=3$ and $r=5$, respectively, without any graph features, produces a prediction accuracy of at least $70\%$. In this case, the baseline achieves only $50\%$, meaning that no synchronizing example is phase-concentrated by a given amount of training iterations. Similarly, for $\dataset{KM}_{15}$, training all four classifiers on only training iterations of $r=5$, without any graph features, produces a prediction accuracy of about $77\%$. In this case, the baseline achieves only $52\%$, so only $10\%$ of all synchronizing examples are phase-concentrated by iteration $r$ (see the formula for baseline accuracy in the section on baseline predictor). This indicates that there may be some evidentiary condition for synchronization on the initial dynamics, which is  \textit{different} from half-circle concentration. A further investigation is due to exactly pin-point what such an evidentiary condition for synchronization might be. }

We give multiple additional remarks on the experiments reported in Figure \ref{fig:full_plot_accuracy}. First, we see that the $\dataset{KM}_{30}$ dataset is adversely affected without the use of the graph features, and beginning at $r=0$, we only achieve 50\% accuracy initially. This is further discussed in the Discussion section along with a systematic analysis of the statistical significance of the features to the classification accuracy in more detail.

Second, the GraphLRCN binary classifier is offset with respect to the other algorithms as it is fed dynamics information encoded into the adjacency matrix of the underlying graph (see eq. (5) in SI), so it cannot be trained only with the initial dynamics information. Oftentimes, GraphLRCN begins below the classical algorithms but is able to outperform them in intermediate iterations, but by the final training iteration, there is only a negligible difference in the performance over different classifiers.

Third, one may be concerned that the few selected graph features we use for training may not give enough information about the given classification tasks, and whether including further graph features may change the results significantly. We remark that GraphLRCN uses the entire graph adjacency matrix as part of the input (see the SI), so technically it uses every possible graph feature during training. In Figure \ref{fig:full_plot_accuracy}, it does show somewhat improved performance on 30 node graphs with the FCA and the GHM dynamics, but virtually identical results for the Kuramoto dynamics. On the other hand, its performance is diminished for 15 node graphs. This may be due to the fact that the training data for 15 node graphs is not rich enough to properly train GraphLRCN, which is a significantly more complex classification model combining convolutional and recurrent neural networks than the other classification models we use.

Fourth, \commHL{in order to investigate the impact of the actual amount of graphs that were used in the datasets with respect to our experiments}, we perform an additional experiment on the 30 node case, in which we subsample 10K and 40K \commHL{balanced datasets} from our 80K sets to see if we get similar performance to Figure \ref{fig:full_plot_accuracy}, \commHL{where the full datasets of 80K examples are used.} As seen in Figure \ref{fig:volume} in the SI, we can see that most methods \commHL{show} a significantly better accuracy \commHL{than} the baseline, with the exception of the \commHL{GB} and \commHL{FFNN} method\commHL{s} for \dataset{KM}$_{30}$ without graph features.
We see however that the \commHL{accuracies between different methods become almost identical as we increase the number of training iterations.}
We note that with only a subsampled portion of the datasets being used,  we see larger differences in accuracy between the methods themselves, which contrasts with Figure \ref{fig:full_plot_accuracy}, where all methods were rather saturated and displayed very similar accuracy across all iterations.

\subsection*{Synchronization prediction accuracy for 300-600 node graphs}

In this section, we investigate Question \ref{Q3}, which is to predict synchronization only based on local information observed on select subgraphs. Note that this is a more difficult task than \ref{Q2}, since not only may we not have information about the underlying graph but we also may not have observed the entire phase configuration. For example, the dynamics may appear to be synchronized at a local scale (e.g., on 30-node connected subgraphs), but there are still large-scale waves being propagated and the global dynamics are not synchronized. Nevertheless, we can use the ensemble prediction method (Algorithm \ref{algorithm:collective_predictor}) to combine decisions based on each subgraph to predict the synchronization of the full graph.

\begin{figure*}[!htpb]
    \centering
     \includegraphics[width=0.85\linewidth]{Figures/figure6.png}

    \caption{Accuracy curves for predicting synchronization of the Kuramoto model on 600-node graphs from dynamics observed from $k\in\{1,2,4,8\}$ subgraphs of 30 nodes. All plots observe the performance of both the ensemble machine learning (solid) and baseline (dashed) accuracies over increasing amounts of training iterations  $r\in\{0,80,240,320,400\}$. The first row shows results using only dynamics whereas the second row includes both the dynamics and graph features. Maximum accuracies for using $k$ subgraphs are given by `$k$-sub: Acc. (Baseline Acc.)'}
    \label{fig:ensemble_plot_accuracy1}
\end{figure*}

In Figures \ref{fig:ensemble_plot_accuracy1} and \ref{fig:ensemble_plot_accuracy2}, we report the synchronization prediction accuracy of the ensemble predictor (Algorithm \ref{algorithm:collective_predictor}) on datasets $\dataset{FCA}_{600}$, $\dataset{FCA}_{600}'$, $\dataset{FCA}_{600}''$, and $\dataset{KM}_{600}$, $\dataset{KM}_{600}'$, $\dataset{KM}_{600}''$, respectively, described in Table \ref{table:datasets_big}. We used Algorithm \ref{algorithm:collective_predictor} with $n_{0}=30$ (amount of nodes in the subgraphs) and $k\in \{1,2,4,8\}$ ($\#$ of subgraphs). The binary classification algorithm we used is FFNN. We chose FFNN because as seen in Figure \ref{fig:full_plot_accuracy}, there is no significant difference in accuracies between all methods used in the 30 node case. Furthermore, we can not use the GraphLRCN model, as \commHL{this method relies on knowing} the dynamics and adjacency matrices of the underlying subgraphs. As the baseline, we use a slight modification of the baseline predictor from the 15-30 node classification task. Namely, we combine all phase values observed on all $k$ subgraphs and predict synchronization if they satisfy the concentration principle otherwise we flip a fair coin. Note that the concentration of phases observed on subgraphs does \textit{not} imply synchronization of the full system, as the full phase configuration may not be concentrated.

\begin{figure*}[!htpb]
    \centering
     \includegraphics[width=0.85\linewidth]{Figures/figure7.png}

    \caption{Accuracy curves for predicting synchronization of 5-color FCA on 600-node graphs from dynamics observed from $k\in\{1,2,4,8\}$ subgraphs of 30 nodes. See the caption of Figure \ref{fig:ensemble_plot_accuracy1} for details.}
    \label{fig:ensemble_plot_accuracy2}
\end{figure*}

So far we have been using the metric of accuracy for our synchronization prediction task, which is defined as the ratio of the correctly predicted synchronizing and non-synchronizing examples to the total number of examples in the dataset. There are multiple ways that this metric could be low, for example, if an algorithm is overly conservative and misses lots of synchronizing examples or in the opposite case it may incorrectly classify lots of non-synchronizing examples. \commHL{In order to provide better insights} on these aspects, we also report the performance of our method (Algorithm \ref{algorithm:collective_predictor}) and the baseline in terms of precision and recall. Here, \textit{precision} is formally defined as the proportion of positive classifications that are correct, or $\frac{TP}{TP+FP}$, where $TP$ is "true positive" and $FP$ is "false positive." In our problem, this corresponds to what proportion of predictions in synchronization was truly correct. \textit{Recall} is formally defined as the proportion of accurately identifying true labels in the data, or $\frac{TP}{TP+FN}$, where $FN$ is "false negative." Again with respect to our prediction problem, measuring recall corresponds to how well a given algorithm, ensemble, or baseline, correctly identifies synchronization behavior when presented with synchronizing data.

Our accuracy results for both the ensemble method and baseline are presented in Figures \ref{fig:ensemble_plot_accuracy1} and  \ref{fig:ensemble_plot_accuracy2} for Kuramoto and FCA models, respectively. In these figures, the first columns represents the results for the datasets $\{\dataset{KM,FCA}\}_{600}$ with fixed number of nodes (600) and relatively smaller variation of edge counts (std $\approx 20, \,38$, respectively); the second columns for the datasets $\{\dataset{KM,FCA}\}'_{600}$ with fixed number of nodes (600) and relatively larger variation of edge counts (std $\approx 80,\, 2371$, respectively); and the third for the datasets $\{\dataset{KM,FCA}\}''_{600}$ with varied number of nodes (std $\approx 86$) and relatively larger variation of edge counts (std $\approx 1461, \,56$, respectively). The first rows of both figures represent prediction accuracies using exclusively dynamics data, and the second row utilizes both dynamics and graph features. In the SI, Figures \ref{fig:ensemble_plot_accuracy3} and \ref{fig:ensemble_plot_accuracy4} show the recall and precision curves of the ensemble and baseline methods with the same row and column orientation as the accuracy figures representing different graph datasets and subsets of features. For the Kuramoto data in Figure \ref{fig:ensemble_plot_accuracy1}, we applied the ensemble and baseline algorithms cumulatively up to training iterations $r=0, 80, 160, 240, 320$ and $400$; and for the FCA data in Figure \ref{fig:ensemble_plot_accuracy2}, we applied the ensemble and baseline algorithms cumulatively up to training iterations $r=0, 10, 20, 30, 40$ and $50$ (Note that using $r=0$ means fitting the prediction algorithms at the initial coloring). We additionally remark that these curves were averaged over 30 train-test splits with 80\% training and  20\% testing.

Across all datasets considered, we see that our ensemble method consistently outperforms the baseline method in accuracy at training iterations $r=400$ for Kuramoto and $r=50$ for FCA. For example, across all Kuramoto datasets and feature subsets by the last iteration, $r=400$, the ensemble method on a single subgraph outperforms the baseline algorithm on eight subgraphs; the best that the baseline algorithm does is 60.42\% accuracy compared to the 91.96\% for the ensemble method's best accuracy. For FCA, the baseline accuracy is 70.29\% compared to the best ensemble method score of 85.79\%, both on eight subgraphs. Considering the recall and the precision plots in Figures \ref{fig:ensemble_plot_accuracy3} and \ref{fig:ensemble_plot_accuracy4}  in the SI gives a more detailed explanation. Namely, the ensemble method significantly outperforms the baseline in the recall by at least 35\%, whereas it performs relatively worse in precision than the baseline by at most $10\%$ for both Kuramoto and FCA with $k=8$ subgraphs except for the datasets $\dataset{KM}_{600}$. This means the baseline is `too conservative' in the sense that it misses correctly classifying a large number of synchronizing examples. From this, we deduce that a large number of synchronizing examples exhibit phase concentration over subgraphs much later in the dynamics---making early detection through phase concentration difficult. On the contrary, the high recall scores of the ensemble methods indicate that our method can still detect most of the synchronizing examples by only using local information observed on the selected subgraphs. To elaborate, the baseline only determines phase concentration at a single point in time, whereas the ensemble method is able to learn the whole variation of dynamics up to iteration $r$. Furthermore, between training our data on dynamics only, versus dynamics \textit{and} graph features, across accuracy, recall, and precision curves, the inclusion of graph features hardly improves the maximum score of these values.

Finally, for both the Kuramoto model and FCA, we observe that high variation of the node and edge counts boosts all performance metrics of accuracy, recall, and precision. For example, for the recall curves we see that for varied edge count, compared to fixed edge count, the recall is considerably higher than fixed edge count by iteration $r=400$, comparing a recall rate of 0.8755 and 0.9759 for 8 subgraphs. Furthermore, the performance gain in introducing a larger variation of node and/or edge counts is significantly larger for the Kuramoto model than for FCA. We speculate that having a larger variation of node and edge counts within the dataset presumably implies a better separation between the synchronizing and the non-synchronizing examples in the space of initial dynamics. We remark that the performance gain here is not due to a better separation between the classes in terms of the graph features, as we can see by comparing the first and the second rows of all figures (Figures \ref{fig:ensemble_plot_accuracy1}, \ref{fig:ensemble_plot_accuracy2} and also Figures \ref{fig:ensemble_plot_accuracy3} and \ref{fig:ensemble_plot_accuracy4}  in the SI).

\iffalse
\begin{figure}[htpb]
    \centering
     \includegraphics[width=1\linewidth]{Figures/600prime.PNG}

    \caption{(Top) Histogram for edge counts of data points in $\dataset{FCA}_{600}'$. (Bottom) Scatterplot of data points in $\dataset{FCA}_{600}''$ with respect to node and edge counts.}
    \label{fig:FCA'_edge_histogram}
\end{figure}
\fi

\iffalse
\begin{figure}[htpb]
    \centering
     \includegraphics[width=1\linewidth]{Figures/histogram_larger_set.png}

    \caption{Empirical distribution of iterations until concentration for the synchronizing examples in datasets $\dataset{FCA}_{600}$ (top), $\dataset{FCA}_{600}'$ (middle), and $\dataset{FCA}_{600}''$ (bottom). }
    \label{fig:iteration_historgram_larger_set}
\end{figure}
\fi

\section*{Discussion}
\label{subsection:gini}

In Figure \ref{fig:full_plot_accuracy}, observe that not using the additional graph features as input (column 4) decreases the prediction accuracy from $70\%$ to $50\%$ for the case of \dataset{KM}$_{30}$ at initial training iteration $r=0$, but there is no such significant difference for \commHL{the discrete models} \dataset{FCA}$_{30}$ and \dataset{GHM}$_{30}$. In fact, in all experiments we perform in this work (those reported in Figures \ref{fig:full_plot_accuracy},  \ref{fig:ensemble_plot_accuracy1}, \ref{fig:ensemble_plot_accuracy2} and Figures \ref{fig:ensemble_plot_accuracy3} and \ref{fig:ensemble_plot_accuracy4}  in the SI), this is only instance that we see that including the graph features during training affects the prediction accuracy.

In order to explain why this is the case, we compare the statistical significance of the all features (including the initial coloring) we use for the prediction task. We do so by computing the Gini indexes of all features by repeating the prediction experiment over 300 train/test splits of the datasets \dataset{KM}$_{30}$, \dataset{FCA}$_{30}$, \dataset{GHM}$_{30}$ using GB for the choice of binary classifier. The analysis using GB will be representative of all other binary classifiers since there are negligible differences in their performance in Figure \ref{fig:full_plot_accuracy}.

As mentioned before, the procedure works by fitting random subsets of features and iteratively growing decision trees; it also records what is known as the Gini index \cite{hartmann1982application} over each consecutive partition through multiple training iterations. As decision trees split the feature space, the Gini index measures total variance across class labels--- in our case, synchronizing v.s. non-synchronizing --- over each partition. Allowing the supposition that synchronization can be modeled through our graph features data using the gradient boosting method, observing the mean decrease in the Gini index across decision trees allows us to infer feature importance for synchronization over different models and graphs (see Figure \ref{fig:giniboxplot}). See, Hartman et al.\cite{hartmann1982application} for more discussions on GB and computing the Gini index.

Interestingly, the discrete models of FCA and GHM place significantly greater importance on color statistics such as the initial quartile colorings, and less importance on graph features such as diameter and minimum and maximum degree. Note that the initial color statistics can be directly computed from the initial dynamics even at training iteration $r=0$, so this explains that there is no significant performance difference in prediction accuracy for the discrete models in Figure \ref{fig:full_plot_accuracy}. On the contrary, the Kuramoto model puts greater importance on graph features such as diameter and number of edges rather than the initial color statistics, and such graph features are not available from the information given by the initial dynamics at training iteration $r=0$. From this, we can see why our algorithms show low prediction accuracy in the case of $\dataset{KM}_{30}$. In addition, we note that there is only a negligible difference in prediction accuracy for $\dataset{KM}_{15}$ in Figure \ref{fig:full_plot_accuracy}. We speculate that this is because for $\dataset{KM}_{15}$, as we see in Table \ref{table:datasets}, having only 15 nodes does not allow to have a significant variation in the diameter and the edge counts of the graphs in the dataset.

Lastly, we remark that the synchronization behavior of the Kuramoto model on complete graphs is well-understood in the literature on dynamical systems
\cite{strogatz2000kuramoto, kuramoto2003chemical, acebron2005kuramoto}. In our experiments, we observed that graphs with small diameters tend to be synchronized, and this aligns with the literature since graphs with the same number of nodes but with smaller diameters are closer to complete graphs.

\begin{figure}[!hbt]
    \centering
     \includegraphics[width=1\linewidth]{Figures/figure8.png}

    \caption{Boxplots of Gini index values sampled from gradient boosting procedure. Color information appears to be very important according to the distribution of Gini index values in both the FCA and GH models, discrete cellular automata models, and diameter overwhelmingly appears to have the greatest importance for Kuramoto.}
    \label{fig:giniboxplot}
\end{figure}

\section*{Conclusion}

\commHL{Predicting whether a given system of coupled oscillators with an underlying arbitrary graph structure will synchronize is a relevant yet analytically intractable problem in a variety of fields. In this work, we offered an alternative approach to this problem by viewing this problem as a binary classification task, where each data point consisting of initial dynamics \commHL{and/or statistics of underlying graphs }needs to be classified into two classes of `synchronizing' and `non-synchronizing' dynamics, depending on whether a given system eventually synchronizes or converges to a non-synchronizing limit cycle. We generated large datasets with non-isomorphic underlying graphs, where classification only using basic graph statistics is challenging. In this setting, we found that pairing a few iterations of the initial dynamics along with the graph statistics as the input to the classification algorithms can lead to significant improvement in accuracy; far exceeding what is known by the half-circle concentration principle in classical oscillator theory. More surprisingly, we found that in almost all such settings, dropping out the basic graph statistics and training our algorithms with only initial dynamics achieves nearly the same accuracy. Finally, we have also shown that our methods are scale well to large underlying graphs by using incomplete initial dynamics only observed on a few small subgraphs . }

Drawing conclusions from our machine learning approaches to the synchronization prediction problem, we pose the following hypotheses:
\begin{description}
    \item[$\bullet$] The entropy of the dynamics of coupled oscillators may decay rapidly in the initial period to the point that the uncertainty of the future behavior from an unknown graph structure becomes not significant.

    \item[$\bullet$] The concentration principle applied at any given time is too conservative in predicting synchronization, and there might be a generalized concentration principle that uses the whole initial dynamics, which our machine learning methods seem to learn implicitly.
\end{description}

Given that our machine learning approach is able to achieve high prediction accuracy, we suspect that there may be some analytically tractable characterizations on graphs paired with corresponding initial dynamics signaling eventual synchronization or not, which we are yet to establish rigorously. As mentioned at the end of the Related Works section, previously known characterizing conditions include the initial vector field on the edges induced by the initial color differential
for the 3-color GHM and CCA \cite{gravner2018limiting}, as well as the number of available states being strictly less than the maximum degree of underlying trees for FCA \cite{lyu2015synchronization, lyu2016phase}. Designing similar target features into datasets and training binary classification algorithms could guide the further analytic discovery of such conditions for the coupled oscillator models considered in this work.

Furthermore, even though we have focused on predicting only two classes of the long-term behavior of complex dynamical systems as only synchronizing and non-synchronizing dynamics, our method can readily be extended to predicting an arbitrary number of classes of long-term behaviors. For instance, one can consider the $\kappa$-state voter model on graphs, where the interest would be the final dominating color. In such circumstances, one can train $\kappa$-state classification machine learning algorithms on datasets of \commHL{non-isomorphic} graphs. \commHL{We can also consider extending our method to be able to predict synchronization on a network based on parameter control. For instance, we can train many different trajectories on a singular graph using different intrinsic frequencies in the Kuramoto model, and learn to predict what range of values of intrinsic frequencies promote synchronization.}

Finally, a more ambitious task beyond long-term dynamic behavior quantified by a single metric is the potential extension of our methods to full time-series and graph state regression. In other words, if each node in the graph represents an individual in an arbitrary social network, can we predict the sentiment level for a given topic at any given time $t$ for every single individual in that particular social network? One can again generate large overarching social networks and run many simulations of sentiment dynamics with many possible edge configurations between individuals (for example, measured by the number of mutual friends or likes/shares of posts on social media). The ultimate goal would be a framework for learning to predict, with precision, entire trajectories of complex dynamical systems.

\section*{Acknowledgments}

This work is supported by NSF grant \href{https://www.nsf.gov/awardsearch/showAward?AWD_ID=2010035&HistoricalAwards=false}{DMS-2010035}. We are also grateful for partial support from and the Department of Mathematics and the Physical Sciences Division at UCLA. We also thank Andrea Bertozzi and Deanna Needell for support and helpful discussions. JV is partially supported by Deanna Needell's grant NSF BIGDATA DMS \#1740325.

\section*{Data Availability}

The codes for the main algorithm used during the current study are available in the repository  \url{https://github.com/richpaulyim/L2PSync}. The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

\small{
	\bibliographystyle{amsalpha}

redpajama_set_name: RedPajamaArXiv


Greenhouse gas (GHG) emissions by constructed wetlands (CWs) could mitigate the environmental benefits of nutrient removal in these man-made ecosystems. We studied the effect of 3 different macrophyte species and artificial aeration on the rates of nitrous oxide (N2O), carbon dioxide (CO2) and methane (CH4) production in CW mesocosms over three seasons. CW emitted 2-10 times more GHG than natural wetlands. Overall, CH4 was the most important GHG emitted in unplanted treatments. Oxygen availability through artificial aeration reduced CH4 fluxes. Plant presence also decreased CH4 fluxes but favoured CO2 production. Nitrous oxide had a minor contribution to global warming potential (GWP < 15%). The introduction of oxygen through artificial aeration combined with plant presence, particularly Typha angustifolia, had the overall best performance among the treatments tested in this study, including lowest GWP, greatest nutrient removal, and best hydraulic properties.


text:
Hospitals at tipping point
Michigan hospitals 'teetering' under current COVID-19 surge
U.S. Army Critical Care Nurse, Cpt. Edward Rauch Jr. (left), tends to a COVID-19 patient on a ventilator at Beaumont Hospital, Dearborn, on Dec. 17. Beaumont is being assisted by 23 military medical personnel from the U.S. Army, sent by the Department of Defense, to assist during the health system's fourth COVID-19 surge. (Photo by JEFF KOWALSKY/AFP via Getty Images)
By Paula Pasche | News-Herald
PUBLISHED: December 22, 2021 at 6:11 a.m. | UPDATED: December 23, 2021 at 10:08 a.m.
Hospitals in southeast Michigan today are in worse shape than they were during the first surge of the COVID-19 pandemic in March 2020.
With the holidays approaching and the unknowns about the omicron variant, hospital administrators are concerned.
"We are bracing for a spike in infections and hospitalizations which could and has the potential to be our largest surge yet,'' Bob Riney, president of health care operations and chief operating officer at Henry Ford Health System, said on Tuesday.
"Our hospitals are teetering and our staffs are past the point of exhaustion, but they're soldiering on like the heroes they are and what you would expect and hope for."
U.S. Army Critical Care Nurse, Cpt. Edward Rauch Jr., prepares to enter a room of a COVID-19 patient on a ventilator at Beaumont Hospital, Dearborn. (Photo by JEFF KOWALSKY/AFP via Getty Images)
As of Tuesday, Henry Ford Health System had 400 COVID patients spread across its five hospitals. That number was down slightly from 440 on Monday.
The situation is similar at Beaumont Health, where on Tuesday, 580 COVID patients packed its eight hospitals.
Veterans answer call to show humanity after Kentucky tornado leaves thousands with nothing
Michigan Lottery warns public to be aware of prize scams
Woman hits $1.6 million Lottery jackpot with ticket sold in Lincoln Park
Federal medical team sent to assist at Wyandotte Hospital as COVID cases surge
Mom said Brownstown salon's refusal to cut crying autistic son's hair heartbreaking
"That number is stable over the last few days, but again still indicating a very significant amount of COVID in our health care system," said Dr. Nick Gilpin, Beaumont Health's medical director of epidemiology and infection prevention. "That's a slight decrease from the numbers we saw a week ago when we were over 600 patients."
About 77% of the hospitalized COVID patients at Beaumont are unvaccinated, officials said. That fits in with numbers from other hospitals around the state.
Dr. John Deledda, chair of the Department of Emergency Medicine at Henry Ford Health Systems, said southeast Michigan is nowhere close to herd immunity and to get to that point, more people have to be vaccinated.
Two major points of concern at Henry Ford — and other hospital systems — include the number of young people infected and a huge increase in staff testing positive.
"In the past week, we've seen a 21% increase in COVID infections in people between the ages of 21 and 35 compared to the previous week," Riney said. "We know that population will be gathering with more vulnerable populations, more vulnerable individuals within their family over the course of the next week. This is also not by coincidence an age group that remains under-vaccinated in a significant number.''
In Michigan, 62.9% of the population has received at least one dose of COVID vaccine, per the Michigan Department of Health and Human Services. But that number drops to 50.34% for ages 20-29.
"We are closely watching the number of our staff who are testing positive from COVID from community spread and we know breakthrough infections are happening at an increasing rate,'' Riney said. "In the last seven days, we had 113 team members test positive, but today we had almost 100 employees test positive for COVID.''
Henry Ford has an employee vaccine mandate so those are breakthrough cases. Even if employees who test positive have mild symptoms, they must remain out of the workforce until they recover and test negative.
"This is an issue in every hospital in Michigan and throughout the country," Riney said. "If we see significant increases in breakthrough infections in our team members that require time off, even if their symptoms are relatively mild, and we are packed to capacity you can see that this will be very, very significant situation as it relates to our ability to staff and take care of our community's health care needs. We are doing so and will continue to do so but we've got to stop the spread of this virus."
Henry Ford has closed 70 of its beds in three hospitals due to staffing shortages. The number — they have a total of 1,500 beds — may not seem significant, but Riney said it's a number that cannot go much higher.
Drive-thru COVID-19 testing site to open in Taylor
COVID deaths up in 2021
New Michigan COVID-19 numbers break daily record
State averages nearly 6,600 new COVID-19 cases per day
Ken Morris: More that ever, it's the time of the season to be charitable
"The COVID-19 surge volume we're seeing in our emergency department is resulting quite frankly in an operating condition that's unsustainable,'' Deledda said. "If we cannot control the spread of COVID-19 infections in our communities — and we do not know yet what omicron is going to bring us — our community's access to safe and high-quality health care is at risk, plain and simple.''
He added that the infrastructure of the health care system — not just at Henry Ford, but across the state — is at a "tipping point."
"This is being caused by an onslaught of preventable COVID-19 infections, the severity of illness that those infections bring and also the subsequent hospital admissions that result from it,'' Deledda said.
Gov. Gretchen Whitmer held a press conference on Tuesday with a similar theme — asking everyone in the state to get vaccinated.
"Getting both vaccine doses plus the booster will prevent serious illness and hospitalization," Riney said. "The research and data continues to confirm that point."
U.S. Army Critical Care Nurse, Cpt. Edward Rauch Jr. (left), tends to a COVID-19 patient on a ventilator at Beaumont Hospital, Dearborn. (Photo by JEFF KOWALSKY/AFP via Getty Images)
Victim confronts suspected phone thief in Allen Park; man threatens him with gun
Four young men suspected of breaking into cars in Southgate
Romulus board president speaks out on recall; organizer prepared to take on members
Alleged drunken driver crashes into optometry office in Allen Park
Six Downriver students named National Merit Scholarship semifinalists
Trenton officers nabbing speeders, other violators in ongoing traffic enforcement sting
Lincoln Park boys' basketball topples Taylor High w/ PHOTO GALLERY
Lincoln Park boys' basketball topples Taylor High w/ PHOTO GALLERY
Romulus fire director warns of 911 abuse after emergency calls made for stubbed toe, toothache
Pastor accused of running his multimillion-dollar Taylor church as a 'cult'
Pastor accused of running his multimillion-dollar Taylor church as a 'cult'
Paula Pasche | Multimedia Journalist
Paula Pasche has worked in a variety of positions, including 32 years as a sports writer, since she joined The Oakland Press. She now covers COVID-related issues and northern Oakland County.
More in Michigan News


As a part of the annual voting process by DJ Mag for the Top 100 Clubs poll which is powered by Miller Genuine Draft, a bonus question was asked to determine the World's #1 Festival and we have a winner!
Ultra Music Festival's flagship, Miami topped the list with almost half a million voters and successfully coming on top of last year's winner, Tomorrowland.
After announcing the win, DJ Mag interviewed members of the Ultra staff; Russell Faibisch, Adam Russakoff, and Charles Faibisch, who are a part of the top tier management at UMF. Check out the interview here.