Episode 11: How to improve to create an optimal structure that is less prone to fragmentation
In
Episode 10, we pointed out that the deep-rooted problem of the 'center-periphery' inherited in wide-area systems, which became apparent when various social infrastructures such as power grids and communication networks collapsed during the Great East Japan Earthquake, also applies to powerful and weak nations, the rich and the poor, and cities and rural areas. The concentration and reliance on hubs in networks can be understood in a similar way. Furthermore, the 14 critical infrastructure sectors include electricity and information and communication, which are considered to have a significant impact on society. Indeed, for example, the massive blackout in North America in August 2003 and the large-scale damage in the Italian peninsula in September of the same year, where the interdependence of power and communication networks led to widespread disruption, have occurred, as have the destruction of social infrastructure due to terrorist attacks and wars in various parts of the world. Behind these events, territorial and economic dominance is undeniably present. Moreover, this is not something we can simply ignore; it has repeatedly affected our daily lives, not only through direct system failures but also through indirect effects such as soaring prices (through imports of crude oil, agricultural products, parts, etc.).
"Today, communications, It is clear that social infrastructure such as logistics and electricity broadly supports our daily economic activities and social life. On the other hand, economic transactions, the movement of people and goods, the supply of electricity, gas and water, information communication via mobile phones and PCs, and the computer networks that control them are all closely interconnected. These are all interconnected and dependent networks (networks of networks), and the role of technological infrastructure in maintaining daily social life can no longer be discussed separately. (omitted) Moreover, this is not a problem of a single network such as electricity or the internet, but rather a more serious problem in that they mutually influence each other and the damage is amplified. (omitted) Natural disasters such as earthquakes, tsunamis, heavy rain and floods, and typhoons occur frequently all over the world, and tend to become increasingly large, possibly due to the effects of the drastic global climate change in recent years. It would not be surprising if sudden torrential downpours or snowstorms occurred anywhere in Japan. Along with such disasters, damage to buildings, land degradation (unsuitable for agriculture and fishing), disruption of logistics, power outages, and disruptions to transportation and communication (due to system failures, etc.) occur. This is causing significant damage." (Quoted from book [11-1], pages 13-16).
As explained in
Episode 9, many real-world scale-free (SF) networks tend to connect to powerful hubs due to a preference for efficiency. However, can you imagine what happens to the connectivity to disasters and attacks when connecting to weaker, lower-order nodes? Recently, it has been scientifically proven that not only does connectivity improve (maintaining connectivity even under greater damage), but the existence of more detours also improves resilience against cascading failures caused by overloads exceeding the processing capacity of each node. The numerous detours correspond to the loop strengthening described later. In
Episode 3,
Episode 4,
Episode 6, and
Episode 7, we explained that a society that eliminates inequality and helps each other is more desirable, and that historically, humanity did so until just a few decades ago. Not abandoning the weak is also the best way to maintain connectivity from a network science perspective. While this is a minor point, when making such comparisons, we assume the same total number of links. This is because, while it's obvious that a larger total number of links makes things less likely to fall apart, the more links there are, the more effort and expense are required for maintenance, making a comparison under the same conditions difficult.
"In preferential attachment, links are added to high-degree nodes corresponding to strong nodes. Conversely, if links are added to low-degree nodes corresponding to weak nodes, how would the degree distribution change? To state the conclusion first, following this inverse preferential attachment reduces the variance (spread) of the degree distribution, decreasing the disparity of degrees. (omitted) Network growth through inverse preferential attachment results in a degree distribution with even smaller variance and a narrower distribution width (smaller disparity) than the exponential distribution, significantly improving connectivity tolerance and communication/transport efficiency." (Quoted from book [11-1], pages 27-28).
"The smaller variance of degree distribution and the closer it is to a random regular graph, the stronger the connectivity tolerance becomes, and the more alternative paths exist." (Quoted from book [11-1], page 127).
If connections are reconfigured from the currently very vulnerable networks, such as power grids and communication networks, to weaker nodes (low-degree nodes) as described above, It has been found that overall connectivity can be better maintained even in the event of disasters or attacks. This means that all nodes have the same degree (number of links) and are equal, approaching a regular graph. Of course, if it is equal, the weak point, the hub, will be eliminated. On the other hand, in terms of efficiency, while the shortest path length (defined as the number of intermediaries + 1) becomes slightly larger in an equal structure, at a realistic scale (total number of nodes from tens of thousands to hundreds of billions), it can be kept to at most double. Therefore, as a principle backed by science, the efficiency-focused network design and construction that has been carried out until now drastically worsens resilience, but the resilience-enhancing approach based on a reverse idea for the near future does not decrease efficiency as much. So, when network growth is not assumed, how can we approach an equal structure? In fact, about 10 years ago, as a hint to this, physicists theoretically demonstrated that the worst-case attack that can cause the network to collapse most haphazardly with the destruction of the fewest nodes is equivalent to node removal that eliminates loops (cycles) from the network and creates a tree structure. Multiple nodes that become loop-free after such removal, this is called the Feedback Vertex Set (FVS) in computer science.
"In recent years, a more fundamental characteristic has been discovered: at critical points where the network is fragmented and destroyed, a fairly wide range of graphs become loop-less, independent of subtle differences in network structure. (omitted) Optimally strengthening attack resistance boils down to finding a network structure that maximizes the FVS size. (omitted) Not only is simply increasing the FVS size important, but increasing the number of nodes essential for forming loops that are not local triangles or quadrilaterals is even more crucial for strengthening connection resistance. Conversely, since a tree structure is always fragmented by removing any node, a structure that is less likely to become a tree even with arbitrary node removal (corresponding to a large minimum FVS size) possesses optimal connection resistance. However, finding the minimum FVS, which is the evaluation metric, itself belongs to an NP-hard combinatorial problem. Therefore, a polynomial-time computation algorithm seems unlikely, and we are forced to rely on approximate solutions." (Quoted from book [11-1], pages 85-86).
Based on the above reasoning, finding the network with optimal resilience would involve measuring the likelihood of various networks becoming tree-like and identifying the one that is least likely to form a tree structure. However, measuring this for even a single network would take an astronomical amount of huge time, making it impossible to solve precisely. Therefore, efforts have been made to find a network that is somehow better, approaching the optimal solution. In particular, increasing the number of loops has proven effective in reducing the likelihood of tree formation, and self-healing methods based on locally distributed processing have even been developed. Here, note that the simplest loop of ring breaks if any link or node is lost, and the loop is strengthened by adding links between nodes belonging to that ring.
"In terms of both robustness, considering resource allocation for a given proportion of nodes, and communication efficiency, self-healing methods based on loop strengthening surpass these conventional methods. While it is not currently clear whether loop formation and subsequent loop strengthening on the loop are optimal for improving robustness, it seems reasonable when considered as a strategy to minimize tree structure and take advantage of the effects of minimum-order node connections." (Quoted from book [11-1], pages 99-100).
To get as close to equality as possible, raising the baseline (to help the weaker nodes) is considered an effective means. Similarly, in networks, to strengthen connection resilience, it has been found that raising the baseline by adding links to the lowest-order nodes during self-organization, as described in
Episode 9, is effective, and that eliminating hubs and creating a completely equal regular graph is ideal. To have optimal resilience, it is necessary to approach equality, but in reality, it functions sufficiently if it approaches to a moderate degree (for example, even if about half of the nodes become dysfunctional, the rest can still connect). However, it has been found that simply approaching equality is not enough, and the loop corresponding to the shortest hole to which each link belongs should not be a triangle or a square, but of a medium length (about log N for the total number of nodes N). Unlike structures such as railway bridges and buildings that primarily maintain their shape, a truss structure made of triangles is rather undesirable for a network whose primary function is transmission and transport. On the other hand, a medium-sized hole, while closely related to the Ramanujan graph, which has been studied for many years in pure mathematics and is considered to possess desirable properties, the way connections are formed does not need to be expressed by rigid mathematical formulas; it is speculated that there is considerable freedom, including the possibility of accidental connections, and its feasibility seems promising. Furthermore, it has recently become clear that in any network that continuously changes from inequality to equality, if there is a modular structure due to the (exclusive and strong) inter-links mentioned in
Episode 9, the resilience of the connections becomes extremely fragile. Also, social infrastructure such as road networks and communication networks are built on the surface of the earth and, for efficiency, follow a planar graph with proximity connections, but it has been found that when power and communication equipment nodes are locally concentrated in urban areas, a modular structure is formed, making them fragile (a different problem of equipment concentration than the population decline in
Episode 10). When tightly united, short loops such as triangles and quadrilaterals tend to form, weakening the resilience of the connections, which is consistent with the properties of holes mentioned above. Therefore, if people become too tightly united with the attitude of 'as long as we're okay', it is easy for division to occur. In fact, scientifically speaking, a looser overall connection actually increases bonding strength.
"The effect of connecting to the lowest-degree nodes is also evident in link-adding strategies that promote the formation of (relatively long-range) global loops." (Quoted from book [11-1], page 94).
"A structure where all nodes have a same degree is the best in terms of connection resilience. The exact opposite of a fragile SF network based on selfish preferential attachment is the direction that future network design and construction should aim for." (Quoted from book [11-1], page 111).
These cutting-edge scientific findings clearly indicate the direction for creating a network with the strongest resistance to disasters and attacks. While the basic concept of resilience (the ability to reconstruction) was explained in the latter half of
Episode 9, we will further supplement this by discussing what constitutes a safe state in a broader sense, maintaining consistency from the perspective of resilience science, such as preventing network fragmentation.
"As a means of maintaining adaptability, in today's world where unpredictable chaos and fluctuations frequently occur, in order to adapt to changing circumstances and achieve one's own objectives, it is necessary to strengthen resistance so as not to be pushed out of a favorable state. Broadening the range of tolerance in preparation for emergencies enhances resilience." (Quoted from book [11-1], pages 88-89).
"In the 21st century, safety must be understood as a state in which things function as well as possible. Therefore, the purpose of safety efforts is not merely to prevent things from going wrong, but rather to ensure that things do function well. This is a new approach to safety, known as Safety-II." (Quoted from book [11-2], page vi).
The before 20th century understanding of 'safety' was passive. Safety-I focused on investigating the cause after an incident occurred and eliminating it as much as possible (within a cost-effective scope, not necessarily complete elimination). This is what is known as post-incident response. Furthermore, it assumed that what could happen was predictable. Even highly advanced technologies, such as predictions by artificial intelligence (AI), fall into this category. The claim that nuclear power plants, mentioned in
Episode 10, are safe is based on this pre-20th-century scientific view and is now completely off the mark.
"Safety-I defines safety as a state in which the number of undesirable events (accidents, malfunctions, near misses, etc.) is kept to a minimum. (omitted) Based on this definition, the objective of safety management is to achieve and maintain the aforementioned state, that is, to reduce the number of malfunction events to an acceptable level." (Quoted from book [11-2], pages 54-55).
"This is the assumption that systems are decomposable, the operation of their components can be described in one of two modes (normal or abnormal), and the order in which events unfold can be precisely determined in advance. (omitted) This idea, valid in the work environment of the early 20th century, has lost its validity today because socio-technological systems are not decomposable, do not conform to two modes, and are not predictable." (Quoted from book [11-2], pages 107-117).
On the other hand, Safety-II, from the 21st century onward, considers the following as a preventative measure: In disasters, even if individual malfunctions are minor and the system functions normally, they often coincidentally combine to cause major malfunctions. It is impossible to anticipate and prepare countermeasures for such a vast number of combinations in advance. Therefore, to enhance network security, it is reasonable to always strive for a structure that closely resembles a regular graph, ensuring connectivity as much as possible. In other words, to ensure that the network is less likely to be fragmented even in the event of a disaster or attack, and that information and supplies can be transmitted and transported effectively, this corresponds to Safety-II. This involves the self-organization and self-repair of the network, as explained above in this Episode 11, (eliminating hubs) while keeping the node order as equal as possible and constantly avoiding internal cohesion (forming short loops such as triangles).
"Safety-II is achieved not by preventing things from going wrong, but rather by ensuring things go well." (Quoted from book [11-2], page 197).
"By analogy with resilience, Safety-II can also be defined as the ability to perform equally well under both anticipated and unexpected circumstances, maximizing the likelihood of intended or acceptable outcomes (in other words, daily activities). (omitted) Both Safety-II safety management and resilience engineering assume that everything happens in essentially the same way, regardless of the outcome. (omitted) From a Safety-II perspective, safety management cannot achieve its objectives through reactive measures alone, because reactive measures only correct what has already happened. To adjust things so that something does not happen, safety management must be proactive. (omitted) This understanding can be deepened by looking for patterns and relationships between events rather than searching for the causes of individual events. To find patterns, it is necessary to spend time understanding what is happening rather than devoting all efforts to finding the causes of individual events." (Quoted from book [11-2], pages 149-154).
The importance of understanding the workings of the world in our daily lives and economic activities as a whole, a network, involving various things including people and organizations (such as companies), can be seen from the following. Unfortunately, as an example of the 'devil' that can occur in many real-world networks, we must understand that there is a tendency (pattern) where the overall connectivity cannot be maintained even if a very small number of hubs become dysfunctional due to disaster or attack, or if internal cohesion becomes too strong. Moreover, this same pattern applies not only to power grids and communication networks, but also to economic systems, ecosystems, and all other networks.
"Using resources to improve Safety-II is not a cost, but rather an investment. (omitted) If we are to explore the perspective of Safety-II, the devil no longer resides in the details, but in the whole. In reality, the devil is not where we can see, but where we cannot see. It is usually ineffective to consider systems, organizations, and socio-technological living environments as a whole and ponder their workings. The usual method of breaking down the subject into small parts does not allow us to understand the whole. Instead, Understanding is only possible by representing the whole as it is." (Quoted from book [11-2], pages 186-187).
To improve such networks, that is, the mechanisms of the world (facing global problems), investment is needed to realize structures with self-repair and optimal resilience, as outlined above in this 11th Episode. In other words, we must transform from the current situation, where many networks are built on prioritizing efficiency and minimizing costs considered wasteful, prioritizing the strong, to a new structure that allows for some resilience and redundancy, connecting with the weak. If we can bring about this transformation, we can create a society where the whole is less likely to be divided, where it can recover even if partially damaged, where social infrastructure can withstand major disasters and terrorist attacks, where reliable connections of mutual assistance among people can be maintained, where inequality is corrected and conflict is suppressed due to equality, and where the global environment and livelihoods are protected.
In the final Episode 12, as a preliminary step towards realizing a network that is less likely to be divided and a society of mutual assistance, we will discuss various measures to achieve this. This concludes this series by emphasizing the need to re-examine our fundamental values.
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