Tag Archives: Interdisciplinary Studies

NEW

  • HOME
  • タグ : Interdisciplinary Studies

How Can Mathematics Help Us to Understand Complex Systems?

February 18, 2016
By 19676

Network analysis has emerged as a key technique in many fields of study, including economics, geography, history, and sociology. One fundamental concept that researchers try to capture is centrality: a quantitative measure revealing the importance of nodes in the network. The values assigned to the nodes are expected to provide a ranking which identifies the most important vertices. Naturally, the word “importance” has a wide range of meanings, leading to many different definitions of centrality. László Csató, a Sylff fellow at Corvinus University of Budapest in Hungary, is exploring the methodological background of some centrality indices.

* * *

In the analysis of complex social and economic structures, actors and the relationships among them are often interpreted as a network. The topology of the network can provide insight into its characteristics and functioning independently from the chosen system (a group of people, a supply chain, international trade relations). The graph-theoretical approach offers a possible approach to modelling these networks. Its strengths lie in the measurement of structural attributes as well as in visualization.

A well-known concept of the graph-theoretic analysis is centrality, which reflects the relative importance of the nodes in the whole network. Many methods exist for this purpose, although not every metric is suitable for every network – the choice depends on the nature of processes in the network and the aims of the analysis.

Most centrality measures have an interpretation on the network graph. However, their axiomatic background deserves more attention: little is known about which measures are excluded and which are supported by accepting a plausible property. The adopted approach is a standard path in (cooperative) game and social choice theory, and is gradually coming to prominence in economics, illustrated by the Nobel Memorial Prize in Economic Sciences awarded to Alvin E. Roth and Lloyd S. Shapley in 2012. It can significantly contribute to the effort to find the right measures for a network.

Since my research has a strong theoretical orientation, I want to illustrate it through an example. Readers interested in the details can consult two working papers by me on the topic.1,2

This paper attempts to develop an index for measuring the accessibility of nodes in networks where each link has a value such that a smaller number is preferred. Examples might include distance, cost, or travel time. In the following, we will give some insight into the results.

Measuring Accessibility

The Marshall Islands in eastern Micronesia are divided into two atoll chains, one of which is Ralik. Figure 1 shows a (simplified) graph of the voyaging network among the 12 islands of the Ralik chain (Ailinglaplap, Bikini, Ebon, Jaluit, Kwajalein, Lae, Lib, Namorik, Namu, Rongelap, Ujae, Wotho). The links between the nodes show the possible routes of inter-island journeys by canoe. For example, it is not feasible to directly travel from Bikini to Jaluit: this journey requires at least five inter-island hops through Rongelap, Kwajalein, Namu and Ailinglaplap. We can therefore say that the distance between Bikini and Jaluit is 5. The problem is to provide a numerical answer to the question “How accessible is a node from other nodes in the Ralik chain?” The islands should be ranked with respect to the probability that they might become the centre of the chain.

An obvious solution is distance sum, the sum of the distances to all the other nodes. For example, Ailinglaplap has a distance sum of 25 since its distance to Bikini is 4, to Ebon 2, to Jaluit 1, and so on. The distance sum is characterized by three independent properties (axioms) such that it satisfies all of them, but any other accessibility index violates at least one property. The three conditions are as follows:

  • Anonymity: The accessibility ranking does not depend on the name of the nodes. Note that Jaluit and Namorik are structurally equivalent in the network, both being connected only to Ailinglaplap and Ebon.
  • Independence of distance distribution: If the distance of a node is decreased and the distance of another node is increased by the same amount, the accessibility ranking does not change.
  • Dominance preservation: A node not far from any other is at least as accessible. For instance, Rongelap should be more accessible than Bikini since the former is closer to certain nodes (e.g., Kwajalein), and there does not exist any island which is farther from Rongelap than from Bikini.

The distance sum focuses exclusively on the shortest paths. Sometimes this is not a desirable feature, as these paths can be vulnerable to link disruptions. Therefore a generalized distance sum, a parametric family of accessibility indices, is suggested. It is linear (easy to calculate), considers the accessibility of vertices besides their distances, and depends on a parameter in order to control its deviation from distance sum. This means that it should violate one axiom of the characterization above, which turns out to be the independence of distance distribution. However, the generalized distance sum is anonymous and satisfies dominance preservation if its parameter meets an appropriate condition.

Figure 2 shows the generalized distance sums for some nodes (on the vertical axis) as a function of a parameter (on the horizontal axis), which measures the influence of the other (i.e., not the shortest) paths. If the parameter is zero, the generalized distance sum is equal to the distance sum, however, some changes can be observed by increasing the value of the parameter:

  • The tie between Ailinglaplap and Wotho (25) is broken for Wotho. This makes sense since the nodes around Wotho have more links among them.
  • The tie between Rongelap and Ujae (27) is broken for Ujae. This is justified by Ujae's direct connection to Lae instead of Bikini as the former is more accessible than the latter.
  • Lae (26), Rongelap, and Ujau gradually overtake Ailinglaplap (25) and Bikini (34) overtakes Jaluit and Namorik (32) in the accessibility ranking. The reason is that the network has essentially two components: the link between Ailinglaplap and Namu is a cut-edge, and the above part around Kwajalein (where Lae, Rongelap, Ujae, and Bikini are located) is bigger and has more internal links.

Kwajalein (with a distance sum of 20) and Namu (21) are the first and second nodes in the accessibility ranking for any value of the parameter. Ebon (34) is “obviously” the least accessible node. These nodes are not shown in Figure 2.

To summarize, the generalized distance sum seems to reflect the vulnerability of accessibility to a disruption on the edge between Ailinglaplap and Namu: if this occurs, then the islands of Ailinglaplap, Ebon, Jaluit, and Namorik suffer more as they have a smaller “internal” network. The parameter can be said to measure this danger to a certain extent.

What Is It Good For?

Accessibility measures can be used in a number of interesting ways:

  • Knowledge of which nodes have the highest accessibility could be of interest in itself (e.g., by revealing their strategic importance);
  • The accessibility of vertices could be statistically correlated to other economic, sociological, or political variables;
  • Accessibility of the same nodes (e.g., urban centers) in different networks (e.g., transportation, infrastructure) could be compared;
  • Proposed changes in a network could be evaluated in terms of their effect on the accessibility of vertices;
  • Networks (e.g., empires) could be compared by their propensity to disintegrate. For example, it may be difficult to manage from a unique center if the most accessible nodes are far from each other.

Network analysis has emerged as a key technique in modern sociology as well as in anthropology, biology, economics, geography, history, and political science. My methodological research aspires to support these applications to get an insight into different networks. I believe robust mathematical foundations are crucial to a better understanding of similar complex systems.


1Csató, L. (2015): Measuring centrality by a generalization of degree. Corvinus Economics Working Papers 2/2015. URL: http://unipub.lib.uni-corvinus.hu/1846/ This paper contributes to network analysis, dealing with the issue of how to identify key nodes in a network. For this purpose a new centrality measure, called the generalized degree, is suggested, based on the idea that a link to a more interconnected node is preferable to a connection to a less central one.
2Csató, L. (2015): Distance-based accessibility indices. Corvinus Economics Working Papers 12/2015.URL: http://unipub.lib.uni-corvinus.hu/1986/

  • HOME
  • タグ : Interdisciplinary Studies

Is There a Link between Music and Language? How Loss of Language Affected the Compositions of Vissarion Shebalin

December 31, 2013
By 19641

How does stroke affect the activities of a musician? Meta Weiss, a cellist and Sylff fellow at The Juilliard School, used an SRA award to conduct research in Moscow about the life and music of Soviet composer Vissarion Shebalin, who lost his linguistic abilities after the second of two severe strokes. By studying Shebalin’s journals and sketchbooks, Weiss gained new insights into the changes in Shebalin’s compositional style after each stroke, which could have broad implications for our understanding of the functioning of the human brain.

* * *

The author performing at the Shebalin Music School

The author performing at the Shebalin Music School

Vissarion Yakevlevich Shebalin was born in 1904 in Omsk, Siberia. He lived in the Soviet Union until his death in 1963 and spent his entire professional life in Moscow. He began his musical studies in Omsk with Mikhail I. Nevitov before transferring to the Peter I. Tchaikovsky Conservatory in Moscow under the tutelage of Nikolai Myaskovsky. Those who knew him always admired his work ethic, modesty, organization, and innate ability as a composer. After completing his studies at the Tchaikovsky Conservatory, he taught there as a professor and eventually became its director, a position which he held from 1942 to 1948.

The year 1948 was a stressful time for all Soviet composers, and Shebalin was no exception. He was accused of formalism and stripped of his position in the Composer’s Union as well as at the Conservatory. Many of his family members believe that the first stroke that he suffered in 1953 was as a result of the many political stresses of the time. He was able to make an almost complete recovery following the stroke in 1953, with the exception of the lingering paralysis of his right arm and leg. He relearned how to write with his left hand, and continued to compose as well as teach. As a teacher, he remained extremely devoted to his students, even during his prolonged illness.

With a former student of Shebalin, Mr. Roman Ledenov

With a former student of Shebalin, Mr. Roman Ledenov

In 1959, he suffered a second stroke that resulted in aphasia. This was especially tragic because of his strong literary background and upbringing; before the strokes he was fluent not only in Russian, but also German, French, Latin and a bit of English. He worked with a team of linguists, neuropsychologists, and doctors to regain the Russian language, and although he was limited in his physical activities by his doctors, he set aside time every day to compose and keep a journal of his activities (with the help of his devoted wife, Alisa Maximovna Shebalina).

By virtue of the fact that Shebalin was a Soviet composer—and he deliberately did not do any self-promotion despite his reputation within the Soviet Union as a leading composer and composition teacher—his music and name essentially died with him in 1963. There is almost no literature on him that is published in English, and when his name does come up in music history articles, it is only in conjunction with the political events of 1948. Shebalin, however, has intrigued the neuroscience community for many years since his case was reported by Drs. Luria, Futer, and Svetkova in the 1960s.

My dissertation will be the first paper in any language to discuss Shebalin’s music through the lens of his medical condition. My aim is to analyze Shebalin’s music, focusing on his string quartets, both pre- and post-aphasia, in order to discover a link, if any, between Shebalin’s loss of verbal language and a change in compositional language. I am collaborating with Dr. Aniruddh Patel at Tufts University, a neuroscientist whose research focuses on music and language. While there have been other (better known) composers who suffered brain injuries, Shebalin is unique in that his condition was characterized by an almost complete loss of verbal language, and we can, based on his sketchbooks, create a fairly accurate timeline of his compositions as well as view the changes in his compositional process. This is inferred by studying the different motivic units Shebalin was constantly writing in his sketchbooks, as well as the more obvious indicators, such as handwriting (left vs. right) and pen color. The string quartets were chosen because they span the creative output of the composer from all periods of his life, and Shebalin himself said that they were the compositions he was most proud of and represented him the best.

After receiving my SRA grant, I traveled to Moscow for the month of October on a student visa and enrolled in the post-graduate program of the Tchaikovsky Conservatory. After exhaustive research working with a Russian-English translator, I was able to track down the location of all of the archival materials on Shebalin, and I was fortunate enough to obtain access to everything that was relevant to my research. This included the RGALI State Archive, the Glinka Museum Archive, and the Tchaikovsky Conservatory Archive and Reading Room.

With Shebalin's family members and directors of the Shebalin Music School

With Shebalin's family members and directors of the Shebalin Music School

Additionally, I was also able, using the Russian “vKontakte” social networking site, to locate the Shebalin family. They granted me permission to visit Shebalin’s summer estate, interview his surviving family members, family friends, former students, and doctors. Two of the three therapists/neuropsychologists who helped Shebalin with his linguistic rehabilitation following his second stroke had already passed away. Again using vKontakte, I reached out to Dr. L.S. Svetkova, the only living team member who treated Shebalin during his rehabilitation, and she agreed to send me her detailed notes and records that she kept while he was her patient.

The highlight of my research was the sketchbooks that are housed in the RGALI State Archive. Shebalin worked quickly and methodically, and was constantly scrutinizing his work. His sketchbooks proved to be much more revealing than any of the manuscripts or other scores. Unlike the detailed journals kept by both Shebalin and his wife, the sketchbooks are unbiased. They show his compositional process neatly and efficiently. Also, because his right side was paralyzed following the first stroke, one can clearly see the change from writing with his right hand to writing with his left hand. They also reveal that after both strokes, he did not simply go back to older works and revise them but he also created completely new and different works.

Shebalin’s music changed in several ways post-stroke. There are distinct differences in the structure of the themes, the imagery of the music, and the scale of his compositions. After his second stroke, he also experimented with a pseudo-twelve-tone style, though still within the tonal idiom, writing themes that featured all twelve tones melodically but relied on the functional harmony of tonality. Perhaps counterintuitively, his music was full of optimism following the onset of his aphasia, and, like his music before the strokes, the music was very clean and straightforward, but with new richness and depth despite the economy of means.

With Shebalin's great-granddaughter and great-great-granddaughter, outside the Shebalin Music School in Moscow

With Shebalin's great-granddaughter and great-great-granddaughter, outside the Shebalin Music School in Moscow

Upon further analysis, it is anticipated that although it will be relatively easy to differentiate the pre- and post-aphasia musical traits, it will be difficult to attribute an exact cause-effect relationship between the change in compositional language and loss of verbal language for two reasons.

First, despite the fact that both Shebalin and his close family and friends—many of whom were interviewed in the course of this research—deny that he ever buckled to political pressure, it will be difficult to definitively separate changes in his music due to political pressure and those due to his medical condition or changing musical taste. The second reason is that because of Shebalin’s fragile physical state following the strokes, he was easily fatigued and thus limited to composing only a few hours a day by his medical doctors. Preliminary analysis reveals that his musical style is markedly more succinct following the strokes, though this may be a result of the doctor’s restrictions.

Through the research conducted with the SRA grant, I was able to construct a complete picture of Shebalin and his compositional output. The future implications of this research are twofold. First, and perhaps most importantly, it would provide hope for stroke victims that in spite of the odds, Shebalin was able to continue to create music and express himself through his most beloved medium—composition. Second, by analyzing his music both pre- and post-aphasia, it may reveal certain processes or elements that are shared by both music and language that remain intact despite the loss of language (such as syntax and grammar).

In this way, my research may help future stroke victims to recover certain aspects of language and aid in our understanding of the brain and mind, a subject with implications far beyond just music or Shebalin.