Figure 9 - available from: Scientific Reports
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The general high-level structure of the US Government 4 . The Enfranchised People occupy the basal, leadership, node. The President's position does not stand out as particularly special in the network structure.
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Coordinated human behaviour takes place within a diverse range of social organisational structures, which can be thought of as power structures with “managers” who influence “subordinates”. A change in policy in one part of the organisation can cause cascades throughout the structure, which may or may not be desirable. As organisations change in si...
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... structure of the US Government is prescribed by the US Constitution, with an intention of creating a system of checks and balances that would prevent a demagogue from capturing too much power and becoming a tyrant 4 . Figure 9 shows the high-level structure of the US Government, with a trophic incoherence parameter of q = 0.4. This is interesting as the structure, as shown, falls into the regime of "tyranny", but with the Enfranchised People in a leadership position, and with the President not holding a particularly special position in the network. ...
Citations
... Finally, extending the model to consider individuals with different importance would be interesting. Some individuals might be more influential because of their status [26], reputation [27], or position in a hierarchical structure [28]. It would be straightforward to account for the importance of each individual when computing group majorities. ...
... Loosely speaking, the hierarchical level ℓ of any node i is defined as the shortest path from a top-node to node i. More precisely, the level ℓ is defined as the trophic hierarchical level 56,57 . We have analyzed the Reddit discussion forum for the Blackberry meme stock below. ...
In social networks, bursts of activity often result from the imitative behavior between interacting agents. The Ising model, along with its variants in the social sciences, serves as a foundational framework to explain these phenomena through its critical properties. We propose an alternative generic mechanism for the emergence of collective exuberance within a broad class of agent-based models. We show that our model does not require the fine-tuning to a critical point, as is commonly done to explain bursts of activity using the Ising model and its variants. Instead, our approach hinges on the intrinsic non-symmetric and hierarchical organization of socio-economic networks. These non-normal networks exhibit transient and unsustainable surges in herd behavior across a wide range of control parameters even in the subcritical regime, thereby eliminating the need for the - arguably artificial - fine-tuning proximity to a critical point. To empirically validate our framework, we examine the behavior of meme stocks and establish a direct linkage between the size of financial bubbles and the degree of non-normality in the network, as quantified by the Kreiss constant. Our proposed mechanism presents an alternative that is more general than prevailing conceptions of instabilities in diverse social systems.
... Trophic analysis was originally derived from ecology [17] where the original definition linked hierarchy to weighted steps from the basal nodes (vertices of in-degree zero), which is an intuitive way to view hierarchy but cannot be generalized to any directed network without basal nodes like the definition used here and in [5][6][7]18]. Much of the previous work which applied trophic level and incoherence used the previous definition [14,17,[19][20][21]. This was successfully applied in a wide variety of settings including infrastructure [22,23], the structure of food webs [20], spreading processes such as epidemics or neurons firing [21] and organizational structuring [19]. ...
... Much of the previous work which applied trophic level and incoherence used the previous definition [14,17,[19][20][21]. This was successfully applied in a wide variety of settings including infrastructure [22,23], the structure of food webs [20], spreading processes such as epidemics or neurons firing [21] and organizational structuring [19]. ...
Knowing which nodes are influential in a complex network and whether the network can be influenced by a small subset of nodes is a key part of network analysis. However, many traditional measures of importance focus on node level information without considering the global network architecture. We use the method of trophic analysis to study directed networks and show that both ‘influence’ and ‘influenceability’ in directed networks depend on the hierarchical structure and the global directionality, as measured by the trophic levels and trophic coherence, respectively. We show that in directed networks trophic hierarchy can explain: the nodes that can reach the most others; where the eigenvector centrality localizes; which nodes shape the behaviour in opinion or oscillator dynamics; and which strategies will be successful in generalized rock–paper–scissors games. We show, moreover, that these phenomena are mediated by the global directionality. We also highlight other structural properties of real networks related to influenceability, such as the pseudospectra, which depend on trophic coherence. These results apply to any directed network and the principles highlighted—that node hierarchy is essential for understanding network influence, mediated by global directionality—are applicable to many real-world dynamics.
... Finally, extending the model to consider individuals with different importance would be interesting. Some individuals might be more influential because of their status [25], reputation [26], or position in a hierarchical structure [27]. It would be straightforward to account for the importance of each individual when computing group majorities. ...
The adaptive voter model allows for studying the interplay between homophily, the tendency of like-minded individuals to attract each other, and social influence, the tendency for connected individuals to influence each other. However, it relies on graphs, and thus, it only considers pairwise interactions. We develop a minimal extension of the adaptive voter model to hypergraphs to study the interactions of groups of arbitrary sizes using a threshold parameter. We study $S$-uniform hypergraphs as initial configurations. With numerical simulations, we find new phenomena not found in the counterpart pairwise models, such as the formation of bands in the magnetization and the lack of an equilibrium state. Finally, we develop an analytical model using a sparse hypergraph approximation that accurately predicts the bands' boundaries and height.
... Trophic Analysis has been used to study many aspects of directed networks, including the structure of food webs (11), spreading processes such as epidemics or signals in neural networks (12), resilience of infrastructure networks (13,14), control of organizations (15), and networks in economics and finance (3). ...
... This will be observed even in random graphs, as trophic incoherence does not usually reach one. The percolation of the strongly connected component and the direction of flow and spread of information may also play a role in communication networks and control and decision-making in organizations (15). ...
In many real, directed networks, the strongly connected component of nodes which are mutually reachable is very small. This does not fit with current theory, based on random graphs, according to which strong connectivity depends on mean degree and degree–degree correlations. And it has important implications for other properties of real networks and the dynamical behavior of many complex systems. We find that strong connectivity depends crucially on the extent to which the network has an overall direction or hierarchical ordering—a property measured by trophic coherence. Using percolation theory, we find the critical point separating weakly and strongly connected regimes and confirm our results on many real-world networks, including ecological, neural, trade, and social networks. We show that the connectivity structure can be disrupted with minimal effort by a targeted attack on edges which run counter to the overall direction. This means that many dynamical processes on networks can depend significantly on a small fraction of edges.
... Overall we propose that the trophic approach can be used as a methodological strategy to explore new patterns and structures for a variety of economic and social science datasets, particularly those with an underlying directed, hierarchical or asymmetrical structure (Pilgrim et al. 2020). Citation networks such as scientific (Zeng et al. 2017), legal (Fowler et al. 2007;Hoadley et al. 2021) and patent (van Raan 2017) networks would be prominent examples. ...
Trophic analysis exposes the underlying hierarchies present in large complex systems. This allows one to use data to diagnose the sources, propagation paths, and basins of influence of shocks or information among variables or agents, which may be utilised to analyse dynamics in social, economic and historical data sets. Often, the analysis of static networks provides an aggregated picture of a dynamical process and explicit temporal information is typically missing or incomplete. Yet, for many networks, particularly historical ones, temporal information is often implicit, for example in the direction of edges in a network. In this paper, we show that the application of trophic analysis allows one to use the network structure to infer temporal information. We demonstrate this on a sociohistorical network derived from the study of hadith, which are narratives about the Prophet Muhammad’s actions and sayings that cite the people that transmitted the narratives from one generation to the next before they were systematically written down. We corroborate the results of the trophic analysis with a partially specified time labelling of a subset of the transmitters. The results correlate in a manner consistent with an observed history of information transmission flowing through the network. Thus, we show that one may reconstruct a temporal structure for a complex network in which information diffuses from one agent to another via social links and thus allows for the reconstruction of an event based temporal network from an aggregated static snapshot. Our paper demonstrates the utility of trophic analysis in revealing novel information from hierarchical structure, thus showing its potential for probing complex systems, particularly those with an inherent asymmetry.
... Loosely speaking, the hierarchical level of any node i is defined as the shortest path from a top-node to node i. More precisely, the level is defined as the trophic hierarchical level [49,50]. We have analyzed the Reddit discussion forum for the Blackberry meme stock (see Section 5 below). ...
We present a generic new mechanism for the emergence of collective exuberance among interacting agents in a general class of Ising-like models that have a long history in social sciences and economics. The mechanism relies on the recognition that socio-economic networks are intrinsically non-symmetric and hierarchically organized, which is represented as a non-normal adjacency matrix. Such non-normal networks lead to transient explosive growth in a generic domain of control parameters, in particular in the subcritical regime. Contrary to previous models, here the coordination of opinions and actions and the associated global macroscopic order do not require the fine-tuning close to a critical point. This is illustrated in the context of financial markets theoretically, numerically via agent-based simulations and empirically through the analysis of so-called meme stocks. It is shown that the size of the bubble is directly controlled through the Kreiss constant which measures the degree of non-normality in the network. This mapping improves conceptually and operationally on existing methods aimed at anticipating critical phase transitions, which do not take into consideration the ubiquitous non-normality of complex system dynamics. Our mechanism thus provides a general alternative to the previous understanding of instabilities in a large class of complex systems, ranging from ecological systems to social opinion dynamics and financial markets.
... This is the definition that will be used in this work. Trophic structure has been used to study spreading processes in neural and epidemiological settings [14], infrastructure [20,21] and the structure of organisations [22]. Trophic Analysis is composed of two parts: the node level information, Trophic Level, which describes where each node sits in the overall hierarchy of a network; and the global information of how directed, or coherent, the overall network is. ...
... This symmetry is broken in a coherent network, as different nodes can have very different abilities to affect the dynamics of the system. The interplay between trophic structure and dynamics has already been observed across a range of systems in the literature [14,18,22]. It has also been shown that the coherence of a network is linked to the non-normality of the adjacency matrix [1,4] (i.e. ...
Many real-world networks are directed, sparse and hierarchical, with a mixture of feed-forward and feedback connections with respect to the hierarchy. Moreover, a small number of 'master' nodes are often able to drive the whole system. We study the dynamics of pattern presentation and recovery on sparse, directed, Hopfield-like neural networks using Trophic Analysis to characterise their hierarchical structure. This is a recent method which quantifies the local position of each node in a hierarchy (trophic level) as well as the global directionality of the network (trophic coherence). We show that even in a recurrent network, the state of the system can be controlled by a small subset of neurons which can be identified by their low trophic levels. We also find that performance at the pattern recovery task can be significantly improved by tuning the trophic coherence and other topological properties of the network. This may explain the relatively sparse and coherent structures observed in the animal brain, and provide insights for improving the architectures of artificial neural networks. Moreover, we expect that the principles we demonstrate here will be relevant for a broad class of system whose underlying network structure is directed and sparse, such as biological, social or financial networks.
... We see in each case a common structural pattern (the same pattern can be observed for most of the empirical networks in Figs. [10][11][12][13][14][15][16][17], where, in particular, a considerably larger fraction of blue and green edges in comparison to red ones indicates that the flow in the ascending hierarchy is considerably higher than that in the opposite direction. The center panels provide an insight into the exact hierarchical structure by indicating the fraction of edges between each hierarchy where we again notice a considerably larger proportion of blue (represented here by the upper triangular elements) and green (those along the diagonal) edges compared to red ones (in the lower triangular part). ...
A large number of complex systems, naturally emerging in various domains, are well described by directed networks, resulting in numerous interesting features that are absent from their undirected counterparts. Among these properties is a strong non-normality, inherited by a strong asymmetry that characterizes such systems and guides their underlying hierarchy. In this work, we consider an extensive collection of empirical networks and analyze their structural properties using information-theoretic tools. A ubiquitous feature is observed amongst such systems as the level of non-normality increases. When the non-normality reaches a given threshold, highly directed substructures aiming towards terminal (sink or source) nodes, denoted here as leaders, spontaneously emerge. Furthermore, the relative number of leader nodes describe the level of anarchy that characterizes the networked systems. Based on the structural analysis, we develop a null model to capture features such as the aforementioned transition in the networks' ensemble. We also demonstrate that the role of leader nodes at the pinnacle of the hierarchy is crucial in driving dynamical processes in these systems. This work paves the way for a deeper understanding of the architecture of empirical complex systems and the processes taking place on them.
... Directionality can also have a crucial effect on dynamical systems. In previous work we have shown, for instance, that trophic coherence is sometimes a determining factor in ecosystem or social stability [9,13], or whether spreading processes such as epidemics will become endemic [14]. And both trophic coherence and non-normality are reflected in graph eigenspectra, which in turn can be related with the stability of dynamical systems [8,10]. ...
Many networks describing complex systems are directed: the interactions between elements are not symmetric. Recent work has shown that these networks can display properties such as trophic coherence or non-normality , which in turn affect stability, percolation and other dynamical features. I show here that these topological properties have a common origin, in that the edges of directed networks can be aligned—or not—with a global direction. And I illustrate how this can lead to rich and unexpected dynamical behaviour even in the simplest of models.
