In real-life (offline) social networks, information diffuses through face-to-face dialogue, a process often referred to as word-of-mouth diffusion. This process is reminiscent of the game of Chinese whispers (or telephone game), where players sit in a circle, and the first person whispers a message to the next. Each participant then whispers what they heard to their neighbor, and so on around the circle. By the time the message returns to the start, it has often transformed dramatically, much to the amusement of all. In a more general context, we call this *information decay*.

In the digital age, however, social media networks facilitate a different mode of information travel. Users often share posts verbatim, replicating content without alteration.

Our new study, published in *Research Policy* introduces a novel measure, *neighborhood centrality*, to quantitatively analyse both kinds of diffusion. The trick is a new parameter that can be interpreted in two ways, depending on the context: as a measure of information decay, or as a probability of verbatim information sharing.

In our setting, which we come back to later, we study the information diffusion through word-of-mouth among academics. One tells the other about a new paper, and might miss something. Thus some share of the information is lost, just like in Chinese whispers.

Existing measures of node importance do not consider information decay, which are fundamental to many diffusion processes. Prominent measures of node importance include *eigenvector centrality**,* *Katz-Bonacich centrality**, *and *diffusion centrality*. These capture node influence in strategic interactions where a pair of nodes can exchange the same piece of information between them ad infinitum. In a previous article, written by Manish Pandey, you can read more about centrality measures and their importance.

What makes neighbourhood centrality widely applicable is a parameter, , which can be interpreted as an information decay parameter in word-of-mouth diffusion processes, and as a probability of information passage in diffusion processes that do not involve face-to-face dialogues (e.g., diffusion in social media platforms). Like in Chinese whispers, a person does not fully understand the issue, or something is lost. That’s what we capture with 𝛿. Alternatively, information decay can be a result of topical distance to the first sender.

Neighbourhood centrality counts the number of all nodes within a given distance from a node, each distance step weighted by to the power of the distance.

Formally, let be the number of -th order neighbours of node . That is, is the number of all nodes at distance (measured as the number of links in the shortest path between a pair of nodes) from . For example, consists of all direct neighbours of , and are all neighbours of ’s direct neighbours.

The neighbourhood centrality, , of node is then the sum over all nodes, discounting for nodes at distance from . That is,

\begin{equation*} N_i(\delta) = \sum_{\tau = 1}^{\infty} \delta^\tau k_{i\tau}. \end{equation*}

In word-of-mouth diffusion processes, is the extent to which the quality of information has decayed by the time it reaches a node at distance from . In diffusion processes that do not involve face-to-face dialogue, and information is shared verbatim, is the probability that a participant shares information with a neighbouring node. The probability that a piece of information seeding from node reaches a node at distance from , is then . In this setting, neighbourhood centrality, , of node is the sum, over all nodes, of the likelihood that a node at a given distance from receives information that seeded from .

Both the intuitive interpretation and the mathematical expression show that neighbourhood centrality extends beyond immediate ties to encompass the broader influence of a node within a network. It considers not only direct neighbours but also neighbours of neighbours, neighbours of neighbour’s neighbours, and so on.

When is small, nodes with the highest degree (direct neighbours) have the highest neighbourhood centrality. As increases towards one, the highest degree nodes will still have a high neighbourhood centrality, but not necessarily the highest. Nodes that are located between high-degree nodes, and those that have the shortest distance to all other nodes, become more important. When , all nodes have identical neighbourhood centrality because regardless of where information seeds from, it will eventually spread to all nodes.

To demonstrate how the ordering of neighbourhood centrality can vary with, consider a network depicted below.

*A simple example of a non-homogeneous network. Nodes with the same colouring have the same number of neighbors.*

The corresponding neighbourhood centralities are computed in the table below.

Node | 0.1 | 0.3 | 0.5 | 0.8 | 0.95 |

1 | 0.12 | 0.52 | 1.30 | 4.25 | 7.48 |

2 | 0.31 | 1.04 | 2.09 | 5.11 | 7.83 |

3 | 0.23 | 0.96 | 2.19 | 5.47 | 7.99 |

4 | 0.23 | 0.96 | 2.19 | 5.47 | 7.99 |

5 | 0.34 | 1.31 | 2.75 | 5.98 | 8.17 |

6 | 0.43 | 1.51 | 2.94 | 6.04 | 8.18 |

7 | 0.13 | 0.66 | 1.72 | 4.99 | 7.82 |

8 | 0.13 | 0.66 | 1.72 | 4.99 | 7.82 |

9 | 0.23 | 0.93 | 2.09 | 5.28 | 7.91 |

10 | 0.11 | 0.49 | 1.30 | 4.39 | 7.56 |

When , the ordering of neighbourhood centrality correlates with the ordering of degree centrality. But as increases, we see a change in ordering. For example, node 2 ranks higher than node 3 when , but for , node 3 ranks higher than node 2. Similarly, when , node 1 ranks higher than node 10, but the order reverses as increases.

Much like the gradual distortion of the message in Chinese whispers, our centrality measure models a diffusion process from a source, where at each step the next node receives only a fraction of the original information. This mimics verbal information diffusion, also in academia. The intuition is that some scientists who engage with a new piece of knowledge at a conference, are among the first to learn about its existence and quality. They then inform their colleagues about it, who then subsequently inform their colleagues, and so on.

We equate the piece of knowledge with an unpublished manuscript, a paper. In our research setting, we just needed an originating source, a seeder. For a researcher to be a seeding node from which information about the existence and quality of the paper can spread to others in the network, they must first familiarise themselves with the paper. We thus focus on discussants who are like co-presenters of a paper on specific academic conferences, except that they are not authors.

Then we assume discussants start a diffusion process. Along this chain of information cascade about the paper, some researchers hear about it. These may, for instance, decide to cite the paper. They may, when they happen to review the manuscript during the subsequent peer-review process, be more familiar with it, which might increase the chances of publication. Both of these outcomes are measurable (number of citations, and the quality of the journal the paper gets published in), and both matter to academics.

We tested the relationship between a discussant’s ability to diffuse information about this paper (i.e., how neighbourhood-central they are) and the annual citation count. We do this in two different academic networks. We tested a network of formal collaboration, where we link authors when they have co-authored a paper together. The other network is a network of informal collaboration, where we link researchers when one acknowledges the other. The contribution of discussants to diffusion of information about the paper is statistically insignificant in both cases. Interestingly, in the formal collaboration networks (where we link researchers when they co-author a paper), the effect is largest at values of between 0.15 and 0.25. This indicates a rather high loss of information or vice-versa a low propensity with which nodes share information.

The neighborhood centrality can be used for more, though. Suppose you are concerned about the spread of fake news on social media platforms. For instance, a doctored video goes viral. How could you limit the spread of fake news?

One possible approach might be to target specific individuals only, namely, those whose position in the network greatly influences the depth of the diffusion process. The question thus becomes: “what is a node’s potential to diffuse information across the network?”

Also here, existing centrality measures fail to capture the fundamental properties of the diffusion process on social media platforms. There information diffuses fundamentally differently. Firstly, information evolves rapidly, and people move on quickly to share new topics and pieces of information. Secondly, the likelihood with which information is reshared (i.e., passed-on to network neighbors) depends on the nature of information, as some information may be more exciting to share, possibly due to its novelty or what it represents. For example, a study published on Science finds that on Twitter, false news stories are 70 percent more likely to be retweeted than true stories.

Neighbourhood centrality captures these properties. Firstly, parameter , which in this case is the probability that a participant shares information with a neighbouring node, is not fixed, and depends on the nature of information being shared (i.e., whether it is fake news, rumors or true stories). Secondly, neighbourhood centrality does not include repeated exchange of a piece of information between pairs of neighbouring nodes, making it a suitable measure of node importance in settings where information evolves rapidly. Finally, neighbourhood centrality considers not only direct neighbours but also neighbours of neighbours, neighbours of neighbour’s neighbours, and so on, and hence, captures a node’s potential to diffuse information across the network!

The featured image is due to Gerd Altmann via Pixabay.

]]>Sounds unbelievable? Well, it's 7:00 am and you just ate breakfast, marking the opening of your gut’s stock market. Each second there are trillions of little traders in your gut, doing all kinds of activities. They have some enemies which they compete with, but there are also some friends who cooperate together, however all are egoistic in the end. Even though they closely resemble the stock traders, their goal is different - instead of fighting for money, the little guys are fighting to replicate as much as possible.

All this hard work in your guts is very worth it - if the process is balanced, it ensures your health and well-being. On the other hand, if their way of life is disrupted, it’s as if the economy crashed. Unhappy little traders can be linked, as researchers have found in various reviews, to many of the modern-day killers: think obesity, diabetes, and even Parkinson’s disease.

These little traders are actually microorganisms and come in all different shapes and sizes, like bacteria, fungi, and even viruses. The community of them living in our gut we call the *gut microbiome*. But nearly every part of life has its own microbiome (collection of different traders): your mouth, hands, fridge, and even your laptop screen! So everywhere you look there’s probably a completely unique community of microorganisms interacting with each other and their surrounding environment.

Now back to our guts, these microorganisms help us digest our food, breaking it down into simple building blocks our bodies can use for energy and other functions. Additionally, these microorganisms help facilitate our immune system, keeping us healthy and stopping unwanted invaders in their tracks. Even our mood is partially determined in our gut, as the majority of the feel-good chemicals like serotonin and dopamine are produced there. Researchers have found for example that the gut provides approximately 95% of total body serotonin, most of which exists in plasma. From this, it should be clear that we want a happy and healthy gut. But what does that actually mean?

In the simplest terms, it means a gut with a diverse range of microorganisms working and competing in a balanced way. But what these interactions look like in the gut is much more complex. So, we need a clever representation to reflect this. This article will show how networks can act as a representation of the microbiome as a population of microorganisms and their interactions.

Researchers studying the microbiome of individuals have already established links between the microorganisms present in samples and specific diseases or illnesses, such as obesity, IBS (irritable bowel syndrome), COVID-19, and even depression.

To figure out what the microorganisms are doing in our gut is very challenging for one key reason - we can’t see the little guys! So we must go about this less directly by looking at what leaves our system. This sample acts as a reflection of the gut microbiome and will be analyzed, via DNA sequencing methods, to find out what microorganisms are present.

The relative amounts of each microorganism can also be determined, which can provide additional insights. We may find that individuals with a certain disease or illness have microbiomes that differ from healthy individuals, their microbiome may be lacking certain microorganisms or may have certain microorganisms that aren’t present in healthy individuals. From large amounts of these studies, we can begin to infer what a healthy microbiome looks like.

This first analysis has only told us what types of little traders are present in our microbiome, but these little guys are not sitting in isolated cubicles, interacting with no one. They are living rich lives full of interactions, fighting with one another for resources, and being part of long chains of production: using someone’s trash to fuel their lives and having their trash fuel another. If we want to get more of the full story, we need to look at the interactions between these microorganisms. However getting this information is difficult - monitoring change requires huge resources and constant measurements. This is where networks come in! Namely, we can construct a network to help us understand the complex interactions between different microorganisms.

To begin constructing such a network, we must infer interactions from the data we’ve already collected. Think of yourself as a private investigator tasked to find out what stock market traders are conspiring together, but the only information you have is pictures of the stock exchange floor on different days. The stock exchange is a bustling place, and each day brings new people making it very difficult to tell who could be conspiring. But you begin to notice a pattern, on certain days there are a few traders who are usually standing near each other. Could they be conspiring or is this due to chance? This is hard to say, but still, you note down the suspicious names, drawing lines between the individuals that always seem to appear next to each other.

Unknowingly, you’ve just created a network. A network is a collection of items, called nodes connected with edges. In this case, the nodes are the stock traders, connected with edges if they are often seen near eacht other. Networks can be weighted if edges have values, or *weights*, associated with the strength of the interaction between two nodes. They can also be directed if the connection between two nodes only runs in one direction.

Microbiome researchers follow a similar line of reasoning as the private investigator. They look at the microorganisms they’ve identified and their relative amounts and identify microorganisms that seem to occur together more often than is likely due to chance. They compile this information into a network where the nodes are microorganisms with edges between them if they often occur together. These edges are weighted to take into account the relative frequency of the two organisms appearing together in a sample. This is called a *co-occurrence network*.

It’s important to note this network gives you no information about actual interactions between microorganisms, but only whether they often occur together. Hence an important question that arises is: how do microorganisms interact with one another? Well, they can cooperate, either by altruistically helping another species or by opportunistically happening to help. Interestingly, for evolutionary stability, there need to be benefits for both sides coming from the cooperation - without it, the collaboration will quickly be discontinued. Microorganisms are very similar to us humans in that regard - no one is entirely selfless. Naturally, there is also a lot of competition. And then there are the truly hated relations, which are entirely one-sided and prey on weaker organisms, like exploitation, predation, and parasitism. In the microbiome, these usually happen when a specific species gets a severe advantage over others in a short time. To get closer to understanding and characterizing the interactions happening right now in our gut we must create a different network, the so-called *interaction network.*

*Figure: Social interactions among microbes. Microbes frequently interact with clonemates or other microbes of the same or different species (represented by microbes of different colors and shapes, respectively). This picture and description were taken from the original article by Alexandre R. T. Figueiredo and Jos Kramer published in Frontiers in Ecology and Evolution.*

To be able to characterize interactions in a general sense we can use information of how the composition of the microbiome changes with time. By collecting samples over days and months we can get this information. Afterwards, we use statistical tools to figure out how the relative proportion of one microorganism changes in response to another. From this, we can infer interactions; cooperation or mutualistic interactions occur if two microorganisms increase in response to one another. Similarly, competitive interactions occur if two microorganisms decrease in response to one another. We can compile this information into another network, and call this an interaction network, signifying the competitive interactions with negative weights and cooperative interactions with positive weights. These networks can be directed if the interaction appears to only be in one direction, for example in a parasite relationship.

Until now we’ve characterized the different types of interactions in our gut, now what? Well, this information about the different types of interactions can tell us a lot about the health of our guts. Similar to the benefits of the daily fights between traders in the stock exchange, organisms in the microbiome should also compete. We want our guts to be filled with a diverse range of mutually beneficial and competitive interactions, a perfect blend of friends, frenemies, and enemies to keep our guts active and our body on its toes. This might be surprising, but in order to stay healthy you want to have a cut-throat atmosphere microbiome in your gut! Why? Well, there are multiple benefits and network science helps us understand them.

Firstly, a study published in Science led to an unexpected conclusion - competition goes hand in hand with network stability - so no more major compositional changes happening in your stomach! Stability is essential in your gut to ensure smooth operations and health. After all, imagine a game of chess between the two best grandmasters in the world. Due to their incredible and unparalleled skill, it is difficult for both to make moves - the game is usually slow and calculated. You can imagine a sort of stability both on the chessboard in a specific game (of course there are minor victories, but usually we cannot speak of a decisive defeat), but also in the leaderboards - most of the best players do not move in rankings that much.

A similar thing happens in your gut with enough competition - obviously, there are some small changes because the system is not static, however, the specific groups of organisms are held at an impasse most of the time.

Let's discuss in more depth the findings in this article about all the costs and benefits of microbial competition. In our gut, microbes fighting for resources are less likely to cooperate. Since microbes are fighting with each other the overall benefits for you are smaller than if they worked together. From a bird's eye view, multiple species fighting for the same resources means that these resources will always be used, and the jobs that are needed will always get done, even as populations change. This is what the paper found, they analyzed the communities (smaller parts) in the network on a local and global level to conclude that too much cooperation destabilizes the microbiome networks and results in difficulty in returning to an equilibrium. Additionally the more diversity in species, the harder it is to achieve stability, as each microorganism has its own unique needs. But this paper found that increasing diversity need not be negative, as long as there is also an increase in competition - stability can then still be achieved.

After discussing that cooperation and competition are both important for stability we will turn to another important question. Namely, what can we learn about the microbiome when observing changes in the interaction networks? What do changes in the microbiome actually mean for the bacteria in your body? Luckily, another article helps us answer these questions. Most strategies that microbes employ are evident in the changes of the network over time, this is the interaction network we talked about earlier - just like in the NYC stock exchange, the exact nature of interactions between actors influences the playing field. So what do these phenomena usually mean? There are three important cases:

**Large changes in node presence and connectivity**: this represents the most basic form of competition (and probably also most common), focusing on fighting for the same resources or trying to eliminate opponents through e.g. toxic antibiotics. Through this mechanism, some microbial populations decrease, whereas others flourish, in turn leading to changes in how important the nodes representing them in the network become.**Community development in the network**: usually caused by indirect effects, groups or communities of microbes working together. Think of this as legal regulators in the stock exchange, influencing how traders work or groups of traders artificially increasing prices to control the market. After all, working in a group produces more impact than working alone. However, these communities can also help elevate the positions of its members in the context of the entire network.**Cycles**: mean that a group of microbes is in a cycle of dominance, without a clear winner - the competition will go on to infinity and beyond because it is almost impossible for a member to break out. Thus, there is no clear winner, instead everyone takes turns being on top.

These are only some examples of competition mechanisms between microbes, but they clearly showcase the strength of using networks in analyzing our guts! Who knows, maybe in the future, they can also help biologists solve many of our health problems.

Having explored some exciting ideas, it is important to reflect on the relevance of the emerging field. The microbiome is not just a craze that a handful of scientists are interested in. It has great implications not only for our health but also understanding of biological systems. After all, 70% of our immune system inhabits our guts. At the same time, microbial communities are complex and largely unexplored.

We’ve seen how to construct two types of networks; co-occurrence networks, from which correlations may be determined, and an interaction network, in which competitive and mutualistic interactions are modeled. From this second network, we can begin to disentangle the different types of competitive mutualistic interactions. If we combine the insights from these networks with in vitro (petri dish) results on certain interactions (say it's known that one bacteria strain produces a product that inhibits its competition from going after the same food), we can get even further in untangling the complex interactions in our gut.

It’s important to understand that the microbiome is so complex that no single technique will be able to map every interaction. Making networks from microbiome samples using statistics gives information on the real-life interactions in the gut, but it is difficult to determine the exact ways in which these microbes interact. Isolating certain microbes and mapping their interactions in a lab gives exact information on how two microorganisms interact, but these interactions may change in the dynamic environment of our guts.

That’s why having a complete picture of our guts is so difficult and has been a booming field of research in recent years. Combining purely statistical methods, like creating co-occurrence networks with biological knowledge, has strong promise. It’s invigorating to imagine what could be possible if we were able to completely map all the interactions in our guts at a given point in time. We could improve our health not by trying to eliminate certain bacteria but by influencing the neighboring microorganisms and regaining stability in our guts. This type of information could completely change the way we treat certain diseases, like obesity, IBS, among many others. Though this future seems very promising we aren’t there yet, and so we must just settle for having a piece of the puzzle, in the form of a co-occurrence or interaction network, and hope with time we will have enough pieces to build the puzzle.

]]>As its name suggests graph theory is the branch of mathematics that studies the properties and structure of *graphs*. Before we discuss the definition, let's start with some examples. Think of the railway network connecting train stations, the network of highways connecting cities, and even your own Google Maps which calculates the fastest route to get you from home to your work.

In general, a network consists of objects with connections between them, and possibly also properties of the objects and their connections. In mathematics, a network is called a *graph*, the objects are called vertices (or nodes), and the connections are called edges. Edges can also have weights, which are numbers that can represent, for example, distances or travel times.

Have a look at the figure on the right. This is an example of a fairly simple graph. We let the numbers represent the distances between cities A, B, C, D, E. A classical question in graph theory is how to compute the shortest route between two nodes in such a weighted graph. To find the shortest route from city A to C, you can calculate the distances of all the routes.

But if we look at even more cities and possible routes between them, we get something like the network in the figure below, where obviously the complexity increases. If we were to calculate all distances separately, we would become utterly miserable. By treating this as a graph theoretical problem, we find that there is a simple algorithm we can use to find the shortest route between cities: Dijkstra’s algorithm.

*The trainrail network of the Randstad in the Netherlands*.

This is one example someone could use to implement graph theory in their lessons. However, the potential of graph theory extends beyond finding shortest routes. Let us look at another topic in graph theory: coloring graphs.

Why would one be interested in coloring graphs and what do we even mean by that? Let us first discuss the why: A map of the world is always displayed with different colors for different countries. But what if we wanted to do this with as few colors as possible? This is where graph theory comes around the corner. In graph theory we can look at the minimum amount of colors needed for coloring a graph such that vertices sharing an edge don't get the same color. This number is called the chromatic number of the graph. Calculating the chromatic number is usually a really hard problem. (On the Network Pages you can find a whole list of articles on this topic.) Yet it’s an engaging challenge that can captivate students’ imagination. We can calculate it for easy graphs like the one at the beginning of the article (you can do this as an exercise!).

But how can we go from maps to graphs? We first identify the countries with nodes. If two countries share a border, they are connected by an edge in the graph. Then we can color the nodes of the graph using the least amount of colors and making sure no vertices sharing an edge get the same color. Afterwards, we color the countries with the corresponding color of the node, so we get something like the figure on the right.

*An example of how to color a map using graph theory.*

We see that we can use graph theoretical problems to show students not all topics in mathematics are solving boring equations. In fact, they can be pathways to understanding and shaping the world around us. And that some parts of mathematics are even applicable to their daily life. So instead of Why do we learn this? we will hopefully get the question: Is there more to learn?.

If you would like more material related to graphs, networks, and algorithms to teach such a subject in your class then you can use the booklets written for the annual masterclasses NETWORKS goes to school!

]]>You come home late in the evening and start cooking your dinner, but to your surprise, you miss the most important ingredient of the recipe. Luckily, nowadays you can order groceries on your phone at the click of a button. Within minutes the groceries are delivered to your doorstep. But how do these companies actually deliver these groceries this fast? How can they ensure that the food is delivered within 10 minutes? To answer these questions we need to take a look at their warehouses.

Gorillas and Flink are two well-known fast-grocery-delivery services in the Netherlands. Both companies make use of a spread of warehouses throughout their regions to maintain short delivery times. So, when your order is delivered too late, you know that the picking process was the cause.

The picking of orders in warehouses is not much different from getting your groceries from the supermarket. You need a collection of items located throughout the building (warehouse or supermarket) and you walk to each item, pick them up, and then leave.

Warehouses, just like supermarkets, consist of a collection of aisles with horizontal aisles at both ends, also called *cross-aisles*. The cross-aisles allow the picker to move from one aisle to another, and can also be placed in the middle of an aisle. Warehouses without such a cross-aisle are called single-block, while those with cross-aisles are multi-block.

When walking into a grocery service warehouse, you would be surprised at how they look incredibly similar to supermarkets but also surprisingly different. The items are stocked similarly and the layout even looks the same. But, the products are no longer grouped by type, instead similar products lay further away from one another. It is more difficult for pickers to see the difference between products that look the same. How quickly can you differentiate between kidney beans and chili beans or garlic and onion powder?

The warehouses of these companies thus take several precautions to prevent long picking times. Nevertheless, the routing of pickers is the most crucial part of warehouse's efficiency, as walking time makes up the largest fraction of the picking time.

How would you walk through a warehouse? Play the game below and try to find the shortest route. You can play the game by clicking the I/O point and dragging your mouse along the aisles. Once you have finished your picking process, you can click the I/O point again to finish the route. You can also use the buttons above to try again, set a new order, or see the optimal route and the so-called S-shaped route.

As you can see the shortest route is quite hard to find, especially in multi-block warehouses! And since each order has different pickup locations, this route can change significantly every time. You can imagine that taking the shortest route, over and over again for different orders, can be quite confusing, as there is no intuition to it. Especially, considering that the warehouse employees don’t see the warehouse from above.

We thus need a smarter way to walk through the warehouse, one that does not confuse the pickers but one that still is sufficiently efficient. For this, we can use the method that most supermarket visitors deploy, the so-called S-shaped routing policy.

*Always looking for the shortest route (left route) can become confusing. This is why the S-route policy is prefered (right route). *

Under this routing policy, you slither through the aisles like a snake, creating several S’s on its side. This implies that the picker walks through every aisle with picks completely and then returns to the input/output point. Click on the S-shaped route button in the puzzle above to see how this works.

But why do we need probability theory?

Each different order that is requested contains completely different items, so each different order needs a different route. A new order has different pick locations than the previous order. Look again at the puzzle above, try pressing new order to see how this changes and see how the routes change as well. Probability theory can now help in analyzing the efficiency of a warehouse. How long does a picker have to walk on average? How can we design our warehouse to make the routes shorter on average?

The pick locations are thus randomly located in the warehouse. The route length of each of these orders thus is different. So how can we still ensure some efficiency?

Consider a warehouse with aisles and no horizontal aisles. Assume that each pick in an order has a probability to be within an arbitrary aisle. So, when a new order, containing a single item, arrives in the warehouse, the probability that an item is requested in aisle 1 is given by .

When an order containing items arrives, this probability becomes . We can see this by rewriting the probabilities as follows:

The random process behind the appointment of picks to aisles is actually related to a classic problem in probability theory: the *occupancy problem*. In this problem, a person throws a fixed number of balls, , at urns. Each different throw has a probability to hit a specific urn. This classic problem, variants of which you might have seen in high school, answers questions like the expected number of urns that are hit or the probability that you hit exactly one urn.

In the order-picking setting, the items can be seen as the balls and the aisles as urns. Every time you throw a ball you reveal an item's location. So, instead of reinventing the wheel, we can use the theory of this seemingly unrelated problem to answer questions about the routes in warehouses. For instance, the expected number of urns that are hit by at least one ball, i.e. the expected number of aisles with at least one pick is given by:

Similarly, the furthest aisle, on average, with picks is the urn with the largest index that has been hit:

Playing around with this formula we see that if we have aisles and orders then on average the furthest aisles with an order are aisles 7, 8, and 9 respectively. For aisles and orders we have that on average the furthest aisles with an order are aisles 20, 23, 28, and 29 respectively. Play around with these formulas and see for yourself what conclusions you can draw about making a warehouse more efficient!

We thus see that a classic problem in probability theory allows us to analyze some crucial elements of the route that pickers take. Namely, the expected distance to the furthest aisle and the expected total number of aisles that have to be visited. Recall that under the S-shaped routing policy, the picker always travels through the entire aisle, so the total length of a route that the picker takes is the sum of the two elements discussed above.

Remark: The exact theory is a bit more involved. When an odd number of aisles has to be visited, the picker ends up on the wrong side of the warehouse, so an extra aisle needs to be visited.

We saw that the routing problem in warehouses has many layers to it, including the choices you make for the routes, but also the randomness in the pick locations for each different order. I showed you how we could apply probability theory to get some insights into the picking time, to see whether or not a routing policy was efficient enough. Hopefully, with this theory, the picker was fast enough to deliver your missing ingredient on time.

]]>The mathematical theory of *probability calculus* was originally developed around 1654 in the correspondences between the French mathematicians Pascal and Fermat to answer certain questions about games of chance and gambling. Nowadays, it is applied successfully in a wide range of topics, many of which are unrelated to games of chance.

The main question of this article What are probabilities? is itself not a mathematical question that can be answered using the probability calculus. Rather it is a philosophical question, investigating for which real world problems the mathematical probability theory can be used.

Probabilities are not restricted to dice, cards or other games of chance. Everyone has to account for uncertainty in their daily lives.

Whenever you go outside, you might consult a weather forecast beforehand to help you decide whether to bring an umbrella or not. If the forecast says there is only a 1% chance of rain, then the extra effort of bringing an umbrella might not be worth the risk of getting wet, while a 50% chance of rain might be sufficient to carry an umbrella around.

*Image from yesofcorsa.com*.

The decisions you make can have far more catastrophic consequences than just getting wet. Traveling by train is safer than traveling by car. So, every time you travel you have to weigh the convenience of traveling by car against its higher chance of having a traffic accident. Though, admittedly, not many people will consciously make a pros-and-cons list of all their travel options. Probabilities also have ethical and moral implications. Judges have to decide how likely it is that a suspect is guilty based on the provided evidence.

A misunderstanding of how probabilities work could lead to erroneous rulings in court. A famous example is the case of Lucia de Berk, who was tried for a number of suspicious deaths that occurred during her hospital shifts. Due to both miscomputation and misinterpretation of the probability that these deaths occurred by chance, she was wrongfully convicted to life imprisonment.

Since probabilities are so pervasive in many aspects of our lives, it might be beneficial to better understand how probabilities work and what they are exactly. While most people will have an intuitive understanding of probabilities, trying to understand what probabilities are and how to interpret them is still an active topic of research in philosophy. This article will not cover the recent developments in this field of research. Rather we will briefly explore the four main schools of thought regarding this subject, and see that none of these gives a completely satisfactory answer to the problem.

The first probability interpretation that we will discuss is the classical interpretation, which is so named as it is the oldest interpretation that was developed. It originated in the 17th century together with the probability calculus itself, but its best known proponent is Laplace with his *Essai philosophique sur les probabilités* from 1814. More recently, in 1968, a variant of this interpretation has been put forward by Jaynes.

This interpretation uses the principle of indifference, which states that if we have a certain number of outcomes and no reason to favor any outcome over the others, then all outcomes have equal probability. The probability of an event is then the ratio between the number of favorable outcomes divided by the total number of outcomes. Here, favorable outcomes are those outcomes for which the event in question happens, not necessarily outcomes that are favorable for you. For example, if we consider the event the die shows an even number, then outcome two is favorable for this event, even if you bet a lot of money on outcome six.

One benefit of this interpretation is that it is easy to use. You might have used this principle to arrive at your answer to the question in the title. A die has six sides and if you don’t have any additional information about he die, then you don’t have any reason to favor one of the sides above any other. So, according to the principle of indifference you should assign probability 1/6 to each of the sides.

However, this interpretation also has some problems. The first problem is that the principle of indifference does not tell us which outcomes to be indifferent about. When applied incorrectly, this principle might give incorrect probabilities. If we throw two dice and consider their sum, then then there are 11 possible outcomes, namely the numbers 2, 3, . . . , 12. Someone with no knowledge of probability calculus might want to apply the principle of indifference to these eleven outcomes and come to the erroneous conclusion that each of these outcomes has probability 1/11. Of course, someone with a better understanding of probabilities will notice that the principle of indifference still works here, but that one should apply it to the 36 different possible outcomes of the two dice. But the indifference principle doesn’t give us any guidelines to specify when it is applicable and when it isn’t.

Another problem is that there are many natural situations in which we would like to be able to assign probabilities to certain outcomes, but in which the indifference principle doesn’t apply. Suppose I have a weighted die that has probability 1/2 to come up six and probability 1/10 to show each of the five other numbers. In order to apply the indifference principle we would have to find ten different outcomes, five of which correspond to the die showing six and one additional outcome for each other number. It seems that such a set of ten outcomes does not exist. Thus, according to the classical interpretation, we simply cannot assign any probabilities to the outcomes of this weighted die, which is an unsatisfactory conclusion.

One way one might try to overcome the difficulties posed by the classical interpretation is to try and test what the probability of an event is. When a random experiment is repeated many times, the fraction of successful trials tends to approach the probability of that event. This fact gives rise to our second interpretation, which is called *frequentism*. According to the frequency interpretation, the probability of an event is the number towards which the fraction of successful trials tends after an infinite number of trials.

The beginnings of this interpretation can be traced back to Jacob Bernoulli’s Ars Conjectandi published in 1713, in which Bernoulli proved the first version of the *law of large numbers*. This mathematical theorem states that, when a *random experiment* (i.e. any situation with a random outcome and not necessarily a science experiment) with a certain success probability is repeated many times, then the frequency of successes will tend to this success probability when the number of trials increases.

For example, during the year 2022, there occured 99436 traffic accidents on Dutch roads that caused material damages or injuries. In the same year, cars traveled a total of 102465.5 million kilometer on Dutch roads (data from CBS). An insurence agent could use this data to say that a person driving one kilometer on a Dutch road has a probability of

to get into an accident.

This interpretation avoids the problems we posed for the classical interpretation. If we want to know what the probability of throwing six is on our weighted die, we simply roll it many times and see towards which number the relative frequency converges.

This interpretation comes with its own set of problems. The most important problem that this interpretation faces, is that it cannot assign meaningful probabilities when the experiment cannot be repeated. While this interpretation can tell you the probability of an arbitrary person having a car crash, it can not assign a probability that you will have a car crash when you decide to drive to work tomorrow morning. The experiment you driving along the highway tomorrow morning cannot be repeated multiple times. So, according to frequentism there is no probability of you having a car crash tomorrow. While the frequency interpretation can be useful for an insurance agent who has to determine how to price their insurance based on traffic accident data, this interpretation might not benefit you when deciding whether to take the train or not.

A second problem for this interpretation is that it requires an infinite number of repetitions of an experiment in order to accurately determine the probabilities of its outcomes. In practice this is obviously not feasible. With more and more repetitions you might be able to get more accurate estimates of these probabilities, but you will never be able to determine them exactly. Furthermore, describing how accurate your estimates of these probabilities are can only be done in terms of probabilities, which leads us to a circular definition.

Like the frequentists, proponents of the propensity interpretation hold the view that probabilities are objective properties of the world. This interpretation was introduced 1910 by Peirce and developed further in 1957 by Popper in an attempt to remedy the problems posed by the frequency interpretation.

According to Peirce, probabilities are tendencies or dispositions of physical systems to behave in a certain way. Such a tendency is a property of a random experiment, similar to the fact that its current velocity is a property of a driving car. Peirce also compares these tendencies to habits that people might have. Since these tendencies are properties of single experiments, this interpretation solves the frequentsists problem that no probabilities can be assigned to nonrepeatable experiments. The fact that frequencies in repeated experiments approach their probability, is no longer their defining property. Instead it is caused by the tendency properties of these experiments.

So, when throwing a fair die, this thrown die will have a tendency of 1/6 to show six, and the strength of this tendency will cause repeated die throws to show six with frequency approximately 1/6. This interpretation is very intuitive and might well correspond with the way in which most people think about probabilities. When I throw a die, I don’t imagine a hypothetical infinite sequence of repeated die throws to get a sense of the probability that I will throw a six, as the frequentists would have me do.

Rather, my intuition tells me that the probability of me rolling a six is a property of this single throw, that does not depend on what would happen if I were to throw the die more often in the future. The problem with the propensity point of view is that it is unclear what these tendencies are themselves, and how to measure the strength of these tendencies. Unlike the velocity of a car, there does not seem to be a way in which we can measure the tendencies of a single die throw. The only way to quantify these tendencies seems to be through repeated experiments. But then this interpretation runs into the same problems as those of the frequency interpretation, which are exactly the problems this interpretation was designed to avoid. Since there is no clear way to measure these tendencies, it also remains unclear what these tendencies are. So, instead of providing an answer to the question what probabilities are, this interpretation only shifts the question away from probabilities towards tendencies.

Furthermore, the propensity interpretation is in contradiction with a deterministic world view. According to this view, the future of the world is completely determined by the current state of all particles in the universe. If this is indeed the case, then nothing in the world is truly random, and all random experiments will have a tendency of 1 to yield a particular outcome. Whether you think this poses a problem for the propensity interpretation, will depend on your views on determinism.

While all interpretations mentioned above agree that there is one correct probability that can be assigned to a given event, the subjectivists (also called Bayesians) take a different stance. According to the subjectivist interpretation probabilities are measures of the degree of belief that someone has that some event will occur. So a single event could have many different probabilities. Each person assigns their own subjective probabilities to that event, depending on their knowledge of the experiment in question.

If you were to roll a fair die, then you are equally convinced that the outcome will be six as that the outcome is any of the other numbers. Hence, your degree of belief in the statement the outcome of the throw will be six is 1/6.

To illustrate how the subjective interpretation differs from the others, let us again consider the weighted die mentioned above, which has probability 1/2 to throw six and probability 1/10 to throw each of the other numbers. If you don’t know that this die is unfair, then your degree of belief that the outcome of a throw with this die will be six is still 1/6. Therefore, according to the subjectivist interpretation, for you the probability of throwing six equals 1/6, while for the conman who is trying to trick you, this probability equals 1/2. This is in contrast with the frequentist and propensity points of view. According to these previous interpretations there is only one correct probability of throwing six with this die, which equals 1/2, and your belief that this probability is 1/6 is simply wrong.

Similarly, while an insurance agent might know the frequency of car accidents among all drivers of your demographic category, you might know that you drive exceptionally careful. Hence, your probability of the event that you cause a car crash might be less than the probability that the insurance agent assigns to this event, which could cause your insurance costs to be higher than you would consider fair.

Not all beliefs behave like mathematical probabilities. Since the mathematical probability calculus seems to work very well, it is desirable that a valid interpretation of probabilities should not be in contradiction with this mathematical theory. In probability calculus, the probabilities of all possible outcomes should add to 1. But in principle, a foolish person might have a belief of 1/2 in each of the six outcomes of a dice throw, even though their subjective probabilities then add to 3.

To remedy this, subjectivists say that only consistent beliefs are probabilities, as it can be shown that whenever someones beliefs are consistent, their beliefs do behave like mathematical probabilities. But again this interpretation has its own difficulties. In practice, most people’s beliefs are not consistent, which is demonstrated for example by various psychological experiments conducted by Kahneman and Tversky. This inconsistency is also illustrated by the following question posed by Allais in 1953.

Consider two different scenarios. In scenario 1 you are given the following choice

- 1A receive a lottery ticket that wins 1 billion euros with probability 11%, and nothing with probability 89%
- 1B receive a lottery ticket that wins 5 billion euros with probability 10%, and nothing with probability 90%.

Which of these two options would you prefer?

In scenario 2 you have two different options

- 2A receive 1 billion euros
- 2B receive a lottery ticket that wins 1 billion euros with probability 89%, nothing with probability 1% and 5 billion euros with probability 10%.

Which of these two other options would you prefer in this scenario?

If, like many other people, you prefer options 1B and 2A, then your beliefs are inconsistent. In scenario 2 you believe that a 1% chance to lose 1 billion euros outweighs a 10% chance to win an additional 4 billion, while in scenario 1 you believe the reverse.

It could also be considered unsatisfactory that this interpretation ascribes multiple probabilities to a single event, based on which information you have on the situation in question. We would like to be able to talk about *the* probability of throwing six with a fair die, not just about *your* probability of doing so.

Which of these interpretations seems most convincing to you? Is one of these interpretations correct? It could be the case that the meaning of the term ‘probability’ differs in different contexts, so the different interpretations could just apply to different situations. Maybe you have your own views on how to interpret probabilities. There is definitely much more to be said on this topic, and likely this article will leave you with many questions to ponder and mull over.

]]>These four posters concern the following topics:

. The poster was created by Stella Kapodistria, assistant professor in the section Stochastics of the Department of Mathematics and Computer Science at the Eindhoven University of Technology, and Peter Verleijsdonk, Doctoral Candidate at the Eindhoven University of Technology.**Dispatching experts to do maintenance**. The poster was created by Anna Priante, assistant professor at the Department of Technology and Operations Management (Business Information Management section) at the Rotterdam School of Management, Erasmus University.**From tweets to communication networks**. The poster was created by Frans de Ruiter who works at the Dutch Data Science Company CQM in Eindhoven.**The Travelling Salesman Problem**. The poster was created by Clara Stegehuis, assistant professor at Twente University.**Structure in social networks**

Habitat fragmentation has led to a loss of as much as 75% of the original biodiversity in certain forests of Australia, the USA, and Brazil (for further reading see this and this article). It has become clear that it is not sustainable to keep expanding our cities without reconstructing part of the occupied habitats. However, this is not an arbitrary task. In order to build habitats where wildlife can thrive, certain conditions ought to be met and proper tools are needed. That is where network science comes into play, bringing insight into the needs of the wildlife living in our cities and helping policy makers assess the effectiveness of their green space proposals.

According to landscape ecology, it’s not. However, in order to understand why, we will need to introduce the concept of *landscape connectivity*. In ecology, landscape connectivity refers to how difficult it is for wildlife to move between different patches with resources. It’s worth noting that how connected a series of green areas is depends on the movement possibilities of the species that are considered. But why is connectivity important?

Well, think of a human colony, no larger than 100 people. Imagine that they are stuck in a small deserted island somewhere in the ocean. Their survival depends on this island providing everything they need. Moreover, if the population continues to grow, or if an event that affects their existing resources takes place, they won’t be able to supplement their demand. Now, if they had access to a boat and there were islands nearby, they could use the resources there to fulfill any needs that were not covered before, as well as possibly increase their consumption.

Translating back to the urban scenario, if the green spaces in our city are parks spread over the city and surrounded by urbanized land, for most species it is effectively the same as living in a deserted island. By constructing paths that connect the different patches, we are providing boats to the wildlife, helping them access different resources and interact with different populations. Therefore, making the ecosystems more resilient and the survival of the different species more plausible.

In the same way that one may build a network out of towns and the roads that connect them, a network of green spaces can be constructed. In this case, the green spaces are the nodes and the paths connecting them are the edges. These paths typically are patches of grass or masses of water like a river, depending on the species of interest and how they move. In our case, we will be dealing with a weighted network, in which the weights of the edges contain information about how hard it is to traverse that path. Such a network contains information on what habitats are available for the wildlife and how easily they can move around these spaces. Factors like distance or habitat suitability of the path influence how easy it is to go from one path to the other. Think again of the human colony on the deserted island. With their boat, they can now travel to the other islands and use their resources. However, if one of the islands was surrounded by strong currents, it may prove almost impossible for them to access it. In our network, the path connecting such an island with their home island would have a high weight; signaling that, although it is connected, traveling there is very hard.

As explained previously, one of the main concerns of the current layout of green spaces of cities is that they are disconnected. To quantify this, we can study the characteristics of our network. For example, the mean total weight of the paths between any two green spaces gives us information on how accessible it is to travel through the current setup. Constructing such a network also allows us to test different configurations helping to decide which paths will lead to better connected green spaces.

There isn’t a singular way to construct the network introduced in the previous section. Since we are considering the urban scenario, in order to assign travel difficulty values, we are mostly estimating which areas are more suitable for our species of interest and are less affected by human industrialization. The closer a patch is to potential danger, the less likely it is to be used by wildlife. Essentially, the main idea is that depending on the danger a threat poses and its range, it increases the travel difficulty of the land around it, and therefore decreases the likeliness of wildlife traveling through it. In the urban setting, industrialized land is considered a threat. Depending on the use of the land, different danger levels and ranges can be assigned. For example, a factory can be more harmful to a larger perimeter of the surrounding environment than a train rail. By considering these threats, we assign travel difficulty values to each piece of land.

Under the assumption that wildlife will take the easiest path between two points, we can compute the routes they are most likely to take between two green spaces and use these to construct our network. We will obtain a network for which all nodes are connected to each other, i.e. a complete network. In order to simplify it, we can consider different procedures. For example, since we assumed that wildlife would take the easiest path, edges directly connecting nodes can be removed if there exists another route between the two green spaces with a lower total travel difficulty. This network contains the information needed to assess the landscape connectivity of our city, helping us determine precisely in what areas improvement is needed.

Achieving a successful transformation to a sustainable urban environment across different cities around the world still remains a big challenge. Barriers dependent on local circumstances like unique social, technical, spatial, and environmental dynamics exist. However, by realizing the need for interventions, the concept of green infrastructure (GI), which considers engineered infrastructure under the context of ecological networks has emerged as the primary tool. For example, China experiencing deterioration within its mega-cities, is looking into GI with multifunctional benefits (the interested reader can have a look here). To overcome some of the challenges faced, the European Commission has been involved in finding and implementing the best strategies for GI since 2013 with the GREEN SURGE project and more recently with project NATURVATION in which diverse cities like Athens and Utrecht are part of.

In the case of Athens, the phenomenon of urban heat island (UHI) has intensified over the years as 80% of the city’s surface turned into built-up areas. A 2018 study identified the city as one of the most vulnerable among 571 European cities against future climate change impacts. Fortunately, Athens secured 5 million in funding from the European Investment Bank under the Natural Capital Finance Facility (NCFF) grant, the following year, to carry out its climate adaption strategies. The long-term plans involved reviving of forests and natural areas, creating open green spaces, and finally creating green corridors between different greened areas. By building the green networks, the city aimed to cater to a multitude of problems and opportunities like creating cool areas, improving air quality, increasing access to green spaces along with the mental health of its residents, preserving biodiversity, and securing a source of income from sustainable tourism.

Innovative methods like installing green roofs and vertical green spaces were also used to build up the green infrastructure (GI). Beyond this, the collaboration between the public-private sector and the local government was paramount to getting to a stage where the summer heat waves were manageable. In 2023, Athens experienced three heatwaves which were well-managed by raising awareness amongst the locals and visitors and having cooling measures in place. The city now aims to continue with its efforts in developing green corridors, implementing nature-based solutions, and rethinking every square meter of public space.

On the northern front of Europe, Utrecht, a city in the Netherlands, pays close attention to landscape defragmentation needs through the Netherlands Nature Network which consists of protected areas and ecological linkages. Any threat to ecological connectivity is solved by ‘the polluter pays’ model. As such is the case of the second largest contiguous reserve in the Netherlands, Utresche Heuvelrug, which was in danger of being fragmented by buildings and roads. Partnerships between the province of Utrecht and municipalities with concerned areas worked together to improve the spatial connectivity of the region. A technical and financially feasible action plan had to be drawn. This step identified 2 corridors that could make the green network more robust. As of now, work on one of the corridors is still underway.

The future of cities lies in prioritizing the preservation of green spaces, accounting for the danger habitat fragmentation poses. By considering the wildlife of our environment as beings with basic needs as us, we can begin to understand where we are failing. Network science can help assess the current situation and serve as a tool for policy makers to determine what changes need to be made in order to make urban areas more suitable for wildlife.

The images in this article where taken from Pixabay.

]]>Before you can start looking for the network communities, we first have to know what a community is. The poster suggests that a community has the following properties:

- There exist relatively many connections between nodes in the same community.

- There exist relatively few connections between nodes in different communities.

To find the network communities, we need a mathematical equation that describes when a division of the network into communities meets these two criteria. One of the first such equations, the so-called *modularity equation*, uses exactly these two community properties.

Suppose we divide a network into several groups. Then the modularity of this network division is an equation with two terms. The first term counts the number of connections within all groups. In short, this says that we would like to have as many connections as possible within the groups. Unfortunately, only maximizing this term will not help.

If we only maximize the number of connections within the groups, then the best division is to put all nodes of the network into one large community. In that case, all network connections are within the same community. To overcome this problem, the modularity equation subtracts a second term from the first term.

This second term describes the expected number of connections within each group if we would redraw the network with the same number of connections per point, but make the connections randomly. Then the division into only one community is not too good anymore. For only one community, we already expected that all connections are within that same community, so the modularity of the one single community is equal to zero. Modularity, therefore, measures how many more connections there are within the communities than we actually expected.

Feel free to print and use this poster for educational purposes. Since this material is protected under a Creative Commons licence we ask you to mention Anna Priante and the Network Pages when using it.

]]>Nowadays surgical operations require advanced robotic equipment. Such equipment can help save lives. Unfortunately, such equipment deteriorates with usage and it can eventually fail. When it fails, it requires maintenance from an expert engineer. Until it is maintained, it cannot be used and hospital operation is disrupted.

Thankfully such equipment is mounted with several sensors that collect data about the condition of the equipment in real time. Using data analytic techniques and Artificial Intelligence, we analyse the data and discover hidden patterns that allow us to often predict (within a margin of accuracy) the failures before they happen. When a failure is predicted, we issue an alert and we plan for preventive maintenance by an expert engineer.

Predicting failures and treating them preventively is cost effective, as maintenance is now planned causing minimal disruption to the hospital operation. Planning the maintenance of the equipment (upon failure or preventively) is a very complicated mathematical optimization problem: At every instant of time, given the available information on the condition of the equipment, we need to decide which engineer to send to treat which issue. However, over time, the available information changes as new data becomes available. So, from one instance of time to the next, as new information becomes available, the problem changes and then we need to solve the problem anew. This makes the problem complicated, but this is not the only complication.

Note that as it stands, the solution to the optimization problem only assigns experts to maintenance issues, but it does not take into account information about the future that is hidden in the data. Analyzing the data, we can often predict (within a margin of accuracy) when an issue will occur. E.g., how long we have in our disposal before an alert is issued or a failure happens. But if we know how long we have in our disposal, we can then strategically reposition the idle experts to a different city so as to ensure they are close to issues when these issues happen. Such repositioning is extremely effective as experts do not wait for an issue to happen but they proactively travel and get close to a city where they will need to perform maintenance in the future. So, by repositioning the experts, we can achieve large coverage with a small response time.

Combining mathematics and Artificial Intelligence, we combine predictions for the future with smart maintenance strategies for the expert engineers. Our solution algorithms consider all available information (current issues and future predictions) and, based on that information, they determine the best way to dispatch and to reposition the experts ensuring as few and as short as possible disruptions at a low cost.

Our poster demonstrates some key insights of the solution algorithm on a small instance of only 5 experts and 18 pieces of equipment. However, keep in mind that in a realistic instance, there are typically hundreds of experts and thousands of equipment. Due to the large number of experts and equipment, it is very challenging to design effective algorithms that can quickly provide good insights. In order to design effective algorithms, we combine knowledge from three mathematical fields: mathematical modeling (we take a real problem and we formulate it into a mathematical problem that captures all its essential elements), data analytics and Artificial Intelligence (we extract hidden patterns from the data), and mathematical optimization (we design algorithms that solve the mathematical problem).

]]>The book covers recent developments in random graphs theory about the local and global structure of random graphs. It is the first book covering the central notion of local convergence, a method that has had profound consequences in random graph theory. Many properties of random graphs can be understood by linking them to the random graph’s local limit.

*The relation between the chapters in Volumes 1 and 2*.

Further, the book covers global properties, such as the existence of a giant connected component containing a positive proportion of the vertices in the graph, as well as when random graphs are fully connected. Another point of focus are the small-world properties of random graphs. The book closes with an extensive discussion on related random graphs models, discussing models that are directed, have a local or global community structure, or have a geometric structure. The book has some 300 exercises that allow readers to sharpen their familiarity with the topics that are discussed. It is online available on my personal website.