text:
Article | Open | Published:

Weighted Betweenness Preferential Attachment: A New Mechanism Explaining Social Network Formation and Evolution

Abstract

The dynamics of social networks is a complex process, as there are many factors which contribute to the formation and evolution of social links. While certain real-world properties are captured by the degree-driven preferential attachment model, it still cannot fully explain social network dynamics. Indeed, important properties such as dynamic community formation, link weight evolution, or degree saturation cannot be completely and simultaneously described by state of the art models. In this paper, we explore the distribution of social network parameters and centralities and argue that node degree is not the main attractor of new social links. Consequently, as node betweenness proves to be paramount to attracting new links – as well as strengthening existing links –, we propose the new Weighted Betweenness Preferential Attachment (WBPA) model, which renders quantitatively robust results on realistic network metrics. Moreover, we support our WBPA model with a socio-psychological interpretation, that offers a deeper understanding of the mechanics behind social network dynamics.

Introduction

Despite the widespread use of the Gaussian distribution in science and technology, many social, biological, and technological networks are better described by a power-law (Zipf) distribution of nodes degree (the node degree is the number of links incident to a node). The Barabasi-Albert (BA) model, based on the degree-driven preferential attachment, generates such scale free networks with a power-law distribution of node degree P(k) = kλ. In fact, degree preferential attachment (DPA) is widely considered to be one of the main factors behind complex network evolution (the scale-free topologies generated with the BA model are able to capture other real-world social network properties such as a low average path length L)1,2. However, recent research challenges the idea that the scale free property is prevalent in complex networks3. Additionally, the degree-driven preferential attachment model has well-known limitations to accurately describe social networks (i.e., complex networks where nodes represent individuals or social agents, and links represent social ties or social relationships), owing to the following considerations:

• People are physically and psychologically limited to a maximum number of real-world friendships; this imposes a saturation limit on node degree4,5. Conversely, in the BA model no such limit exists.

• People have weighted relationships, i.e., not all ties are equally important: an average person knows roughly 350 persons, can actively befriend no more than 150 people (Dunbar’s number)4, and has only a few very strong social ties (links)6. The BA model does not account for such link weights7.

• The structure and dynamics of communities in social networks are not accurately described with DPA7,8,9,10,11.

To address these issues, recent research has combined the DPA model with properties derived directly from empirical data. For instance, there exist proposals which add the small-world property to scale-free models (e.g., Holme-Kim model12, evolving scale-free networks13) or the power-law distribution to small-worlds (e.g., the Watts-Strogatz model with degree distribution14, multistage random growing small-worlds15, evolving small-worlds16, random connectivity small-worlds17). Other research proposals extend Milgram’s experiment18, e.g., static-geographic19 and cellular20 models. However, all these models are still not accurate enough when compared against real-world social networks.

To better understand the real-world accuracy problem, we perform a topological analysis on a variety of real-world network datasets and show that node betweenness (which expresses the node quality of being “in between” communities) is power-law distributed and–at the same time–correlated with link weight distributions. Our empirical findings align well with previous research in some particular cases11,21. Such empirical pieces of evidence suggest that, for social networks, the node degree is not the main driver of preferential attachment; therefore other centralities may be better attractors of social ties. We conclude that node betweenness–as opposed to node degree or any other centrality metric–is the key attractor for new social ties.

Consequently, as the main theoretical contribution, we introduce the new Weighted Betweenness Preferential Attachment (WBPA) model, which is a simple yet fundamental mechanism to replicate real-world social networks topologies more accurately than other state-of-the-art models. More precisely, we show that the WBPA model is the first social network model that is able to replicate community structure while it simultaneously: (i) explains how link weights evolve, and (ii) reproduces the natural saturation of degree in hub nodes.

Finally, we further interpret WBPA from a socio-psychological perspective, which may explain why node betweenness is such an important factor behind social network formation and evolution.

Results

Centrality statistics

We investigate the distributions of node betweenness on a variety of social network datasets: Facebook users (590 nodes), Google Plus users (638 nodes), weighted co-authorships in network science (1589 nodes), weighted online social network (1899 nodes), weighted Bitcoin web of trust (5881 nodes), unweighted Wikipedia votes (7115 nodes), weighted scientific collaboration network (7343 nodes), unweighted Condensed Matter collaborations (23 K nodes), weighted MathOverflow user interactions (25 K nodes), unweighted HEP citations (28 K nodes), POK social network (29 K nodes), unweighted email interaction (37 K nodes), IMDB actors (48 K nodes), Brightkite OSN users (58 K nodes), Facebook - New Orleans (64 K nodes), respectively Epinions (76 K nodes), Slashdot (82 K nodes) and Timik (364 K nodes) on-line platforms. To improve the robustness of our analysis, we ensure data diversity by considering network datasets with different sizes, weighted and unweighted, and representing various types of social relationships (see Methods).

Our first observation is that, in all datasets, node degree, node betweenness, link betweenness, and link weights (for datasets with weighted links) are power-law distributed. Moreover, the power-law slope of degree distribution is steeper in comparison with node betweenness distribution. More precisely, as presented in Fig. 1a, the average degree slope is γdeg = 2.097 (standard deviation σ = 0.774) and the average betweenness slope is γbtw = 1.609 (σ = 0.431), meaning that γdeg is typically 30.3% steeper than γbtw across all datasets (details in SI.1. Social network datasets statistics). Also, for all considered datasets there is a significant non-linear (polynomial or exponential) correlation between node betweenness and node degree (see Fig. 1b); this further suggests that node betweenness may be the source of imbalance in node degree distribution. The statistics for the entire dataset collection are presented in SI.1.

The second observation is that–unlike node degree–node betweenness is significantly more correlated with the weights of the incident links. After assessing the correlation between both node betweenness and node degree with the weighted sum of all adjacent links, we argue that betweenness acts as an attractor for stronger ties. For example, for the co-authorships weighted network with 1589 nodes23, the top 5% links accumulate 27.4% of the total weight in the graph; these top 5% links are incident to nodes which amass 80.2% of the total node betweenness, but only 14.9% of the total node degree (see Fig. 2–further numerical details in SI.1, Table 2). In all analyzed weighted datasets, node betweenness correlates with incident link weights by ratios that are 2.5–9 times higher than node degree–link weights associations (additional details in SI.1, Fig. 2).

The first observation indicates a significant correlation between node degree and node betweenness but it does not necessarily imply causation. However, the second observation is that betweenness attracts stronger links which, in turn, triggers more imbalance in degree distribution; this suggests that node betweenness is behind networks evolution, while the power-law degree distribution is only a by-product. The importance of node betweenness is further supported by the analysis of centrality dynamics. To this end, we provide the example of an on-line social network, UPT.social, which was intended to facilitate social interaction between students and members of faculty at University Politehnica of Timişoara, Romania24. Right after its launch in 2016, UPT.social attracted hundreds of users, and the entire dynamical process of new links formation was recorded as snapshots of the first 6 weeks (T0 − T5). As exemplified in Fig. 3 (and further detailed in SI.3, Fig. 6), the nodes with high betweenness become the principal attractors of new social ties; we also note that the top 3 nodes attracting new edges at time snapshot T2 are the ones which maximize their betweenness beforehand, and then trigger a subsequent degree increase. As shown, once node degree begins to saturate (T3 − T5), node betweenness drops, as nodes fulfill their initial bridging potential.

Betweenness preferential attachment (BPA)

In what follows, we propose the betweenness preferential attachment model (BPA) and conjecture that–for social networks–it is more realistic than the degree preferential attachment (DPA) model. The fundamental difference between the degree-driven and betweenness-driven preferential attachment is illustrated in Fig. 4; the upper panel shows that, under the DPA rule, the nodes with high degree (colored in orange) gain an even higher degree. In contrast, the lower panel in Fig. 4 shows that, under the BPA rule, the nodes with high betweenness (orange) attract more links and increase their degrees; in turn this decreases their betweenness via a redistribution process, thus limiting the number of new links for high-degree nodes as a second order effect. This may explain why, in real-world networks, the number of new links is limited for high degree nodes (i.e., degree saturation).

WBPA model

Besides validating the BPA mechanism, we also realize that all the empirical network data gathered in a real-world context is weighted, even if the information about link weights is not always available. For example, there is no link weight information in our Facebook and Google Plus datasets, yet these networks are clearly part of a weighted social context in which each link has a distinct social strength. Realistic networks evolve according to a mechanism which considers link weights, therefore we develop the weighted BPA (WBPA) algorithm to characterize the social network evolution.

The WBPA algorithm for link weight assignment according to the fitness-weight correlation is given in Fig. 5 and discussed below. In the case of WBPA, the fitness f is node betweenness. Note that even though link weights wij are not used directly during the growth phase, they have a significant second order impact: Betweenness depends on the shortest paths in the graph, which in turn are highly dependent on link weights. Link weights are updated in step 3 of the WBPA algorithm, and whenever a weight becomes ≤0, the corresponding link is removed.

Weighted BPA Algorithm (WBPA)

1. 1)

Distribute weights: Begin with an arbitrarily connected graph G with nodes V and bidirectional links E (i.e., for eijeji). A weight wij is added for each link eij in the graph, so that wij is proportional to fitness fj of the target node vj. For each node vi, all incident link weights wij are normalized so that the outgoing weighted degree is 1.

2. 2)

Growth (BPA): At every step, a new node vk is introduced; the new node tries to connect to n (1 ≤ n ≤ V) existing nodes in G. The probability pi that vk becomes connected to an existing node vi is proportional to fitness fi. Therefore, we have $${p}_{i}={f}_{i}/{\sum }_{j\in V}\,{f}_{j}$$ where the sum is made over all nodes in the graph.

3. 3)

Dynamic weight redistribution: Once a new node vk becomes connected to an existing node vi, weights wki and wik are initialized with the normalized fitnesses fi and fk respectively. As the weighted outgoing degree of node vi increases by wik, every other weight wij is rescaled with −wik/n, where n is the previous number of neighbors of node vi.

Assessing the realism of WBPA

WBPA defines complex interactions between link weights and node centralities, hence we expect emerging phenomena such as n-order effects. Therefore, a mathematical analysis of WBPA would be cumbersome and beyond the scope of our paper. Instead, as validation strategy, we test WBPA against several preferential attachment (PA) models to explore which one produces the most realistic social network topology. To this end, we quantify preferential attachment according to a fitness function f which expresses the capability of individual nodes to attract new connections (e.g., if f is chosen to be node degree Deg, then we reproduce the classic BA model2). We consider f as one of the following network centralities: degree Deg (DPA model), betweenness Btw (WBPA model), eigenvector centrality EC (ECPA model), closeness Cls (ClsPA model), and clustering coefficient CC (CCPA model). Each node centrality is defined in the Methods section. The comparison between synthetic and real-world networks is done through topological similarity assessment supported by the statistical fidelity metric25, alongside standard deviation and p-values. Fidelity takes values φ [0, 1] with 1 representing a network that is identical with the reference network (see the Methods section for more details).

We also make use of the following graph metrics to characterize and compare networks: average degree (AD), average path length (APL), average clustering coefficient (ACC), modularity (Mod), graph diameter (Dmt), and graph density (Dns). We start by measuring the distributions of these six metrics on the 18 selected real-word datasets. To assess which centrality is the most appropriate as fitness function, we start by generating networks according to each PA model, of increasing sizes: N = {1K, 2K, 5K, 10K, 50K, 100K} nodes; the full statistical results are presented in SI.2. Best fitness for preferential attachment. Aggregating the statistical results from SI.2–Fig. 4 (real-world data) and Fig. 5 (PA networks), we provide an intuitive visual comparison in Fig. 6 between the averaged evolution of the six graph metrics on the real-world data (N = 590 to N = 364 K nodes), and on the degree-driven and betweenness-driven PA networks.

To better illustrate the comparisons between the synthetic PA networks and the real-world datasets, we present the trend lines for each graph metric in Fig. 6; for the real-world data networks the trend line is green-dotted, for Btw fitness networks is blue, and for Deg fitness networks is red. On close inspection, we uncover the following:

• AD in real data evolves differently than in PA networks.

• APL evolution in real data resembles Btw networks much better than Deg networks. We measure a statistical fidelity of φBtw = 0.925 and φDeg = 0.853.

• ACC evolution in real data resembles Btw more than Deg, with statistical fidelities of φBtw = 0.665 and φDeg = 0.515.

• Mod evolution in real data resembles both networks very well, with statistical fidelities of φBtw = 0.814 and φDeg = 0.812 (a slight advantage for the Btw networks).

• Dmt evolution in real data resembles Deg more than Btw. Even though we see the same type of increase, Deg produces longer diameters as seen in the majority of real-word data. The measured statistical fidelities are φBtw = 0.796 and φDeg = 0.836.

• Dns evolution in real data resembles both networks, with statistical fidelities of φBtw = 0.634 and φDeg = 0.634.

For simplicity, Fig. 6 includes only Deg and Btw PA networks in the comparison with real-world data; the full numerical data–with all PA network models–are detailed in Table 1. All these results demonstrate the superior realism provided by the WBPA in comparison to the classic DPA principle, as well as in comparison to PA driven by other node centralities such as eigenvector, closeness or clustering coefficient.

We strengthen our analysis by presenting several direct comparisons between real networks and synthetic PA networks, generated with the same node sizes as the real-world reference networks. The comparisons are made using the fidelity metric φ, as well as by comparing individual graph metrics (one by one), to show that WBPA is superior to the other PA networks. To this end, we select the Facebook (FB), Google Plus (GP), Online social network (OSN), and IMDB real-world datasets, and provide the full statistical results in Table 2; here, each sub-table contains the reference real-world network and its graph metrics on the first row, while the remaining lines contain the averaged graph metrics for 10 synthetic networks generated according to preferential attachment driven by each centrality (Deg, Btw, EC, Cls, CC). Additionally, we provide measurements for a Null model (Random network) to serve as baseline. The standard deviation for each synthetic dataset metric is symbolized with a ± sign.

The mechanism of preferential attachment which we adopt in our paper is a fundamental, yet generic and simple framework. State of the art studies which are specifically aimed at creating realistic topologies propose algorithms with a far increased complexity. Therefore, intuitively, it is expected that state of the art models like Cellular (Cell)20, Home-Kim (HK)12, Toivonen (TV)26, or Watts-Strogatz with degree distribution (WSDD)14 etc., will generate more realistic topologies in terms of the six discussed graph metrics. To test this hypothesis, we further generate such synthetic networks of size N = 10,000 and compare them with WBPA, DPA networks and several real-world datasets. The results are provided in Table 3, showing that not only is WBPA superior to DPA and PA models driven by other centralities but, in most cases (i.e., 10 out of 13), it outperforms the other synthetic models in terms of topological fidelity as well. For readability purposes we did not add information about the standard deviations of each synthetic model here; this information may be found in SI.4, Tables 4 and 5.

To offer the diversity required by a robust test of our model, we also include unweighted networks in our collection. A fair comparison between WBPA networks (which are all weighted) and the large and unweighted example networks, requires that all weights on our WBPA algorithm output be discarded. In this comparison, we start by generating WBPA networks of 10,000 nodes, then make all weights $${w}_{ji} > 0$$ become 1, thus obtaining unweighted BPA networks.

The upper half of Table 3 contains the average fidelities of WBPA, DPA and the two null model networks, towards the real-world reference networks. The lower half of Table 3 contains the other state of the art synthetic networks. Our WBPA obtains the highest fidelity towards most empirical references, e.g., 13–68% higher φFB, 21–81% higher φOSN, 4–47% higher φTK than all other synthetic models. As such, we prove the increased realism of our model in comparison with some elaborated state-of-the-art models (briefly described in SI.4, and quantified in SI.4, Table 4). Compared to DPA, our model produces networks with higher fidelity values; when averaged over all empirical networks we obtain: $${\overline{\varphi }}_{Btw}=0.831$$ and $${\overline{\varphi }}_{Deg}=0.777$$.

We note that the WBPA model produces a specific distribution of the Betweenness/Degree (B/D) ratio. To this end, we measure B/D distributions on all datasets (weighted and unweighted), as well as on our synthetic WBPA-generated networks, using the Gini coefficient (a Gini coefficient takes values between 0 and 1, with values closer to 0 representing a more uniform dispersion of data) to evaluate data dispersion27. The Gini values obtained on the empirical data are given in Table 4: all empirical datasets, whether weighted or unweighted, have their Gini coefficients within a similar range, i.e., the average real-world Gini is greal = 0.5193 ± 0.071. Indeed, for WBPA networks with 10,000 nodes, we have an average Gini coefficient of gWBPA = 0.4962 ± 0.0282, which is very close to the real-world B/D Gini values (−4.5%). Additionally, we generate 10 of each random, small world, and PA networks of 10,000 nodes. For these synthetic networks we obtain the corresponding Gini values in Table 4. The PA networks (except WBPA) produce an average gPA = 0.7784 ± 0.0128, whereas the random network produces an average Gini grand = 0.9374 ± 0.0013. These results point out two key aspects: (i) the B/D dispersion in other PA and other state-of-the-art synthetic models differs significantly from real-world social networks, and (ii) WBPA produces networks with B/D distributions that are closer to the real-world.

Two specific B/D distributions are exemplified in Fig. 7a,b for the Google Plus and POK users networks, respectively. Figure 7c,d present the B/D distribution for the DPA and WBPA networks. The visual similarity inspection reveals WBPA as the only synthetic model capable of reproducing the real-world B/D ratios (see SI.1, Fig. 3 for additional examples).

The WBPA realism is also backed up by the centrality distribution analysis. The power-law slopes for degree and betweenness distributions in WBPA (γdeg = 1.391 and γbtw = 1.171) are very similar to the real-world distributions from the Centrality statistics section (see Fig. 1) and SI.1, Table 1, meaning that the degree slope is steeper than the betweenness slope (with 18.8%). Similar to the real-world cases, we obtain a polynomial fit for the node betweenness-degree correlation in WBPA (y = 0.246x2 + 329.8x − 3569.4, with correlation coefficient R2 = 0.9977).

Discussion and a Socio-Psychological Interpretation

From a computational standpoint, node betweenness is significantly more complex to compute in comparison with node degree. However, when individuals make assessments of social attractiveness in real-world situations–which is essential for driving preferential attachment and establishing new social links–they do not rely on executing algorithms or other types of quantitative evaluations. Instead, individuals make decisions based on qualitative perceptions30. In light of the quality over quantity hypothesis proposed by social psychology31, we argue that node betweenness is a far better indicator of social attractiveness than node degree, because the quality of being “in between” can be easily and quickly perceived, due to the fact that humans are better at observing qualitative aspects (e.g., differences and diversity) than quantitative ones32. This idea is supported by an experimental study on how people favor investing in fewer qualitative social ties, rather than numerous lower quality ties32. Our results indicate that WBPA provides a more accurate social network topological model, being able to reproduce real-world community structure as well as to explain degree saturation and link weight evolution.

We believe that the WBPA model transcends the mere topological perspective on social relationships evolution. As such, in the field of social psychology, individuals are perceived as social creatures who strive for social recognition, validation, approval and fame7,19,33,34. Indeed, individuals tend to connect to two types of other nodes: individuals who are popular in their communities (i.e., typically they have high degree), and individuals who connect multiple communities (having high betweenness). While the former type of interconnection is mostly related to the popularity of individuals within local communities, it appears to be an epiphenomenon of the latter.

Also, state of the art has previously identified that social networks have apparent (degree) assortative mixing, while, technological and biological networks appear to be disassortative in nature34,35. The study in35 explains this as most networks have a tendency to evolve, unless otherwise constrained, towards their maximum entropy state–which is usually disassortative. A similar debate was introduced by Borondo et al. based on the concepts of meritocracy versus topocracy36. The authors discuss the critical point at which social value changes from being based on personal merit, to being based on social position, status, and acquaintances. In the context of social networks, we interpret this issue as follows: in our ego-networks the balance between friends with less influence and ones with more influence than us translates into betweenness assortativity. Indeed, connecting to persons with high betweenness and increasing our tie strength with them (through, say, a stable social relationship), we ourselves become, in turn, more influential social bridges. This propagation of influence determines other persons, with lower betweenness, to interact with us and direct more tie strength towards us.

Towards this end, we introduce the concept of social evolution cycle, which revolves around betweenness assortativity rather than degree assortativity34,35,37. According to our approach, individuals become more influential over time by increasing their own betweenness. Therefore, the exhibition of one individual’s desire to increase his/her betweenness is two-fold: it attracts new ties (i.e., increase in degree), and it creates stronger ties (i.e., increase in link weight); this process continues for the next generation of individuals who aspire to climb the social ladder. As shown, this conclusion is supported by the evolution of networks generated with WBPA.

We envision two ways of improving an individual’s social status. The first choice relies on forcing tie strengths inside the existing neighborhood to increase first, followed by an increase in influence. The second choice relies on increasing influence first by broadening the neighborhood to influential agents (BPA principle), which will in turn trigger an increase in tie strengths. We consider the second choice as the more plausible social process, as detailed and explained in Fig. 8.

We conclude that the WBPA model is quantitatively more robust than DPA, as it can reproduce more accurately a wide range of real-world social networks. Such a conclusion means that node degree is not the main driver in social network dynamics. Instead, node betweenness is a much better indicator of social attractiveness, because it drives the formation of new social bonds, as well as the evolution of social status of individuals. From a socio-psychological standpoint, individuals (intuitively) perceive node’s betweenness as the capacity of bridging communities, irrespective of its degree. As shown, WBPA is a subtle mechanism at work that is able to replicate the social network community structure. Also, WBPA explains the dynamic accumulation of degree and link weights, as well as the eventual degree saturation, as a second order effect. Consequently, we believe our work paves the way for a new and deeper understanding of the mechanisms that lie behind the dynamics of complex social networks.

Methods

Real-world datasets

All data used in this study were selected to facilitate a thorough analysis of node betweenness and degree, as well as measuring the realism of synthetic networks. The real-world datasets have been chosen based on diversity of both context and network size. Prior studies confirm that data mining from sources such as Facebook or Google Plus is reliable for realistic social network research38,39, and indicate a strong correlation between the real-world and virtual friendships of people40,41.

Table 5 provides the graph metric measurements used for the realism assessment of our WBPA model, as presented in the Results section. Our real-world datasets comprise the following social networks (ordered by network size, from N = 590 to N = 364K nodes): Facebook (FB) users41, Google Plus (GP) users28, weighted co-authorships (CoAu) in network science23, weighted on-line social network (OSN)22, trade network using Bitcoin OTC platform (BTC)42, votes for Wikipedia administrators (WkV)43, weighted scientific collaboration network in Computational Geometry (Geom)44, Condensed Matter collaboration network from arXiv (CM)45, weighted interactions on the stack exchange web site MathOverflow (MOvr)46, High-Energy Physics citation network (HEP)47, POK online social network29, Enron email (EmE) communication network48, IMDB adult actors co-appearances, Brightkite online social network (BK)49, Facebook-New Orleans (FBNO)50, Epinions online social network (EP)51, Slashdot online social network (SL)48, and Timik online platform (TK)52.

Information about the nature of nodes and links, as well as direct URLs for each dataset are provided in SI.5 Datasets availability, Table 6. In the main manuscript, Table 6 presents the natural ranges for the graph metrics that are provided in Table 5, as they are measured across the entire range of considered real-world on-line social networks41.

Network centralities

All graphs are generated and visualized using Gephi53; the graph centralities are analyzed using the poweRlaw package distributed with R according to the methodology described in54. Full details for the topological analysis of data are given in SI.1. Furthermore, to quantify the specific distributions of B/D ratios introduced in this paper we made use of the Gini coefficient–borrowed from the area of economics where it is used to evaluate data dispersion27.

In SI.2 we present the preferential attachment analysis based on combinations of two and three node centralities. Given a graph G = (V, E), with nodes viV and links eijE, we define the basic graph centralities and metrics used throughout the paper. We represent the adjacency matrix as W = {wij}, which contains either the weight of the link for any link eij, or 0, if no link exists. If the network is unweighted, then each wij = 1.

The degree ki of a node vi (also denoted as D) is defined as $${k}_{i}=\sum {w}_{ij}$$. In case of directed networks, there is a differentiation between in-degree and out-degree, but that is beyond the scope of this subsection. The average degree AD of the graph is calculated over all nodes as1:

$$AD=\frac{1}{n}\sum _{i\in G}{k}_{i}$$
(1)

The clustering coefficient CCi measures the fraction of existing links in the vicinity Vi of a node, and is formally defined as55:

$$C{C}_{i}=\frac{|\{{e}_{jk}\,|\,j,\,k\in {V}_{i}\}|}{{k}_{i}({k}_{i}-\mathrm{1)}}$$
(2)

with ki being the degree of node vi, and ejk the set of links connecting two friends in the vicinity of node vi, all divided by the maximum number of links in vicinity Vi. Consequently, the average clustering coefficient ACC of the entire graph is the average of all CCi over all nodes.

Considering d(vi, vj) as the shortest path between two nodes in G, the average path length APL is defined as1:

$$APL=\frac{1}{n(n-\mathrm{1)}}\sum _{i\ne j\in G}d({v}_{i},{v}_{j})$$
(3)

If there is no path between two nodes, then that particular distance is considered 0; n is the total number of nodes |V| in G.

The diameter of a graph is defined as the longest geodesic56, namely the longest shortest distance between any two nodes: Dmt = max(d(vi, vj)).

Graph density is simply defined as the ratio between number of links and maximum possible number of links, if the graph were complete56. For undirected graphs, it is defined as:

$$Dns=\frac{\mathrm{2|}E|}{n(n-\mathrm{1)}}$$
(4)

Modularity is a measure for quantifying the strength of division of a graph into modules, or clusters, and is often used in detection of community structure57. Modularity Mod is the fraction of the links which lie within a given group minus the expected fraction if links were distributed at random. Values for Mod range between [−1/2, 1). If it is positive, then the number of links within a cluster exceeds the expected number. Also, a high overall modularity means dense connections between the nodes within modules and sparse connections between nodes in different modules. We use the algorithm of Blondel et al. to compute modularity58.

Betweenness centrality is commonly defined as the fraction of shortest paths between all node pairs that pass through a node of interest1, and is defined as59:

$$Btw({v}_{i})=\sum _{i\ne j\ne k\in G}\frac{{\sigma }_{jk}({v}_{i})}{{\sigma }_{jk}}$$
(5)

where σjk(vi) is the number of shortest paths in G which pass through node vi, and σjk is the total number of shortest paths between all pairs of two nodes vj and vk from G.

Closeness centrality is defined as the inverse of the sum of geodesic distances to all other nodes in G1,56, and can be considered as a measure of how long it will take to spread information from a given node to other reachable nodes in the network:

$$Cls({v}_{i})={(\sum _{{v}_{j}\in G\backslash {v}_{i}}d({v}_{i},{v}_{j}))}^{-1}$$
(6)

where d(vi, vj) is the distance (number of hops) between the two nodes vi and vj.

The most common centrality based on the random walk process is the Eigenvector centrality (EC), which assumes that the influence of a node is not only determined by the number of its neighbors, but also by the influence of each neighbor23. The centrality of any node is proportional to the sum of neighboring centralities1. Considering a constant λ, the EC is formally defined as:

$$EC({v}_{i})=\frac{1}{\lambda }\sum _{{v}_{j}\in {V}_{i}}EC({v}_{j})$$
(7)

Assessing network fidelity

In order to assess the structural realism of the generated social networks, we used the statistical fidelity φ, which is proven to offer reliable insights on complex network topologies25. The fidelity metric φ numerically captures the similarity between any graph topology G* with respect to another reference graph G (i.e., a complex network G = (V, E)). More precisely, by measuring and comparing their common individual graph metrics, a maximum fidelity of 1 represents complete similarity, while a minimum fidelity of 0 represents complete dissimilarity between the two compared topologies. Of note, the fidelity is not dependent on the choice of metrics of interest, however it is customizable to allow a weighted comparison. Depending on the context of the problem, any numerical value (i.e. metric) that is representative for the model can be used. The definition and proof of statistical fidelity φ are detailed in25.

Definition 1. Given a reference topology G, and any other network G* being compared to G, the arithmetic fidelity $${\phi }_{A}^{\ast }$$, which expresses the similarity between G* and G, is defined as:

$${{\phi }}_{A}^{\ast }=\{\begin{array}{ll}\frac{1}{n}\,\sum _{i=1}^{n}\,\frac{{m}_{i}}{2{m}_{i}-{m}_{i}^{\ast }} & if\,{m}_{i}^{\ast } < {m}_{i},\,{m}_{i}=0\\ \frac{1}{n}\,\sum _{i=1}^{n}\,\frac{{m}_{i}}{{m}_{i}^{\ast }} & if\,{m}_{i}^{\ast }\ge {m}_{i},\,{m}_{i}=0\\ \frac{1}{n}\,\sum _{i=1}^{n}\,\frac{1}{{m}_{i}^{\ast }+1} & if\,{m}_{i}=0\end{array}$$
(8)

In equation 8, i is the index of the metric which describes the two networks being compared, and n is the total number of metrics used in the comparison. In this paper we compute the fidelity between multiple synthetic topologies and the empirical social network references. These reference datasets are chosen because they have typical real-life social network features. The fidelity comparison is made relative to the set of relevant network metrics (indexed by i).

In this paper, fidelity is measured by taking into consideration the following topological characteristics: average degree AD, average path length APL, average clustering coefficient ACC, modularity Mod, diameter Dmt, and density Dns.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. 1.

Wang, X. F. & Chen, G. Complex networks: small-world, scale-free and beyond. Circuits and Systems Magazine, IEEE 3, 6–20 (2003).

2. 2.

Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. science 286, 509–512 (1999).

3. 3.

Broido, A. D. & Clauset, A. Scale-free networks are rare. arXiv preprint arXiv:1801.03400 (2018).

4. 4.

Dunbar, R. I. Neocortex size as a constraint on group size in primates. Journal of Human Evolution 22, 469–493 (1992).

5. 5.

Brashears, M. E. Humans use compression heuristics to improve the recall of social networks. Scientific reports 3 (2013).

6. 6.

Krackhardt, D. The strength of strong ties: The importance of philos in organizations. Networks and organizations: Structure, form, and action 216, 239 (1992).

7. 7.

Adamic, L., Buyukkokten, O. & Adar, E. A social network caught in the web. First monday 8 (2003).

8. 8.

Strogatz, S. H. Exploring complex networks. Nature 410, 268–276 (2001).

9. 9.

Newman, M. Networks: an introduction (Oxford University Press, 2009).

10. 10.

Burt, R. S. Attachment, decay, and social network. Journal of Organizational Behavior 22, 619–643 (2001).

11. 11.

Abbasi, A., Hossain, L. & Leydesdorff, L. Betweenness centrality as a driver of preferential attachment in the evolution of research collaboration networks. Journal of Informetrics 6, 403–412 (2012).

12. 12.

Holme, P. & Kim, B. J. Growing scale-free networks with tunable clustering. Physical Review E 65, 026107 (2002).

13. 13.

Fu, P. & Liao, K. An evolving scale-free network with large clustering coefficient. In Control, Automation, Robotics and Vision, 2006. ICARCV'06. 9th International Conference on, 1–4 (IEEE, 2006).

14. 14.

Chen, Y., Zhang, L. & Huang, J. The watts–strogatz network model developed by including degree distribution: theory and computer simulation. Journal of Physics A: Mathematical and Theoretical 40, 8237 (2007).

15. 15.

Jian-Guo, L., Yan-Zhong, D. & Zhong-Tuo, W. Multistage random growing small-world networks with power-law degree distribution. Chinese Physics Letters 23, 746 (2006).

16. 16.

Wang, J. & Rong, L. Evolving small-world networks based on the modified ba model. In Computer Science and Information Technology, 2008. ICCSIT'08. International Conference on, 143–146 (IEEE, 2008).

17. 17.

Zaidi, F. Small world networks and clustered small world networks with random connectivity. Social Network Analysis and Mining 1–13 (2013).

18. 18.

Milgram, S. The small world problem. Psychology today 2, 60–67 (1967).

19. 19.

Lazer, D. et al. Life in the network: the coming age of computational social science. Science (New York, NY) 323, 721 (2009).

20. 20.

Tsvetovat, M. & Carley, K. M. Generation of realistic social network datasets for testing of analysis and simulation tools. Tech. Rep. DTIC Document (2005).

21. 21.

Leydesdorff, L. Betweenness centrality as an indicator of the interdisciplinarity of scientific journals. Journal of the American Society for Information Science and Technology 58, 1303–1319 (2007).

22. 22.

Opsahl, T. & Panzarasa, P. Clustering in weighted networks. Social networks 31, 155–163 (2009).

23. 23.

Newman, M. E. Finding community structure in networks using the eigenvectors of matrices. Physical review E 74, 036104 (2006).

24. 24.

Topirceanu, A., Garcia, J. & Udrescu, M. Upt. social: The growth of a new online social network. In Network Intelligence Conference (ENIC), 2016 Third European, 9–16 (IEEE, 2016).

25. 25.

Topirceanu, A. & Udrescu, M. Statistical fidelity: a tool to quantify the similarity between multi-variable entities with application in complex networks. International Journal of Computer Mathematics 94, 1787–1805 (2017).

26. 26.

Toivonen, R., Onnela, J.-P., Saramäki, J., Hyvönen, J. & Kaski, K. A model for social networks. Physica A: Statistical Mechanics and its Applications 371, 851–860 (2006).

27. 27.

Xie, Y. & Zhou, X. Income inequality in today’s china. Proceedings of the National Academy of Sciences 111, 6928–6933 (2014).

28. 28.

McAuley, J. J. & Leskovec, J. Learning to discover social circles in ego networks. NIPS 2012, 548–56 (2012).

29. 29.

Takac, L. & Zabovsky, M. Data analysis in public social networks. In International Scientific Conference and International Workshop Present Day Trends of Innovations, 1–6 (2012).

30. 30.

Tversky, A. & Kahneman, D. Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty 5, 297–323 (1992).

31. 31.

Rowatt, W. C., Nesselroade, K., Beggan, J. K. & Allison, S. T. Perceptions of brainstorming in groups: The quality over quantity hypothesis. The Journal of Creative Behavior 31, 131–150 (1997).

32. 32.

Shirado, H., Fu, F., Fowler, J. H. & Christakis, N. A. Quality versus quantity of social ties in experimental cooperative networks. Nature communications 4, 2814 (2013).

33. 33.

Plous, S. The psychology of judgment and decision making. (Mcgraw-Hill Book Company, 1993).

34. 34.

McPherson, M., Smith-Lovin, L. & Cook, J. M. Birds of a feather: Homophily in social networks. Annual review of sociology 415–444 (2001).

35. 35.

Johnson, S., Torres, J. J., Marro, J. & Munoz, M. A. Entropic origin of disassortativity in complex networks. Physical review letters 104, 108702 (2010).

36. 36.

Borondo, J., Borondo, F., Rodriguez-Sickert, C. & Hidalgo, C. To each according to its degree: The meritocracy and topocracy of embedded markets. Scientific reports 4 (2014).

37. 37.

Zhou, D., Stanley, H. E., DAgostino, G. & Scala, A. Assortativity decreases the robustness of interdependent networks. Physical Review E 86, 066103 (2012).

38. 38.

Hossmann, T., Legendre, F., Nomikos, G. & Spyropoulos, T. Stumbl: Using facebook to collect rich datasets for opportunistic networking research. In World of Wireless, Mobile and Multimedia Networks (WoWMoM), 2011 IEEE International Symposium on a, 1–6 (IEEE, 2011).

39. 39.

Ferrara, E. & Fiumara, G. Topological features of online social networks. arXiv preprint arXiv:1202.0331 (2012).

40. 40.

Valenzuela, S., Park, N. & Kee, K. F. Is there social capital in a social network site?: Facebook use and college students’ life satisfaction, trust, and participation1. Journal of Computer-Mediated Communication 14, 875–901 (2009).

41. 41.

Topirceanu, A., Udrescu, M. & Vladutiu, M. Genetically optimized realistic social network topology inspired by facebook. In Online Social Media Analysis and Visualization, 163–179 (Springer, 2014).

42. 42.

Kumar, S., Spezzano, F., Subrahmanian, V. & Faloutsos, C. Edge weight prediction in weighted signed networks. In Data Mining (ICDM), 2016 IEEE 16th International Conference on, 221–230 (IEEE, 2016).

43. 43.

Leskovec, J., Huttenlocher, D. & Kleinberg, J. Signed networks in social media. In Proceedings of the SIGCHI conference on human factors in computing systems, 1361–1370 (ACM, 2010).

44. 44.

Batagelj, V. & Mrvar, A. Pajek-program for large network analysis. Connections 21, 47–57 (1998).

45. 45.

Leskovec, J., Kleinberg, J. & Faloutsos, C. Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data (TKDD) 1, 2 (2007).

46. 46.

Paranjape, A., Benson, A. R. & Leskovec, J. Motifs in temporal networks. In Proceedings of the Tenth ACM International Conference on Web Search and Data Mining, 601–610 (ACM, 2017).

47. 47.

Leskovec, J., Kleinberg, J. & Faloutsos, C. Graphs over time: densification laws, shrinking diameters and possible explanations. In Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining, 177–187 (ACM, 2005).

48. 48.

Leskovec, J., Lang, K. J., Dasgupta, A. & Mahoney, M. W. Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics 6, 29–123 (2009).

49. 49.

Cho, E., Myers, S. A. & Leskovec, J. Friendship and mobility: user movement in location-based social networks. In Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, 1082–1090 (ACM, 2011).

50. 50.

Viswanath, B., Mislove, A., Cha, M. & Gummadi, K. P. On the evolution of user interaction in facebook. In Proceedings of the 2nd ACM workshop on Online social networks, 37–42 (ACM, 2009).

51. 51.

Richardson, M., Agrawal, R. & Domingos, P. Trust management for the semantic web. In The Semantic Web-ISWC2003, 351–368 (Springer, 2003).

52. 52.

Jankowski, J., Michalski, R. & Bródka, P. A multilayer network dataset of interaction and influence spreading in a virtual world. Scientific Data 4, sdata2017144 (2017).

53. 53.

Bastian, M., Heymann, S. & Jacomy, M. Gephi: an open source software for exploring and manipulating networks. In ICWSM (2009).

54. 54.

Gillespie, C. S. Fitting heavy tailed distributions: the powerlaw package. arXiv preprint arXiv:1407.3492 (2014).

55. 55.

Watts, D. J. & Strogatz, S. H. Collective dynamics of small-world networks. Nature 393, 440–442 (1998).

56. 56.

Newman, M., Barabasi, A.-L. & Watts, D. J. The structure and dynamics of networks (Princeton University Press, 2011).

57. 57.

Newman, M. E. Modularity and community structure in networks. Proceedings of the National Academy of Sciences 103, 8577–8582 (2006).

58. 58.

Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. Journal of statistical mechanics: theory and experiment 2008, P10008 (2008).

59. 59.

Newman, M. E. The structure and function of complex networks. SIAM review 45, 167–256 (2003).

Author information

Affiliations

1. Department of Computer and Information Technology, Politehnica University of Timişoara, Timişoara, 300223, Romania

• Alexandru Topirceanu
•  & Mihai Udrescu

Contributions

A.T., M.U., and R.M. designed research, analyzed data and wrote the paper; A.T. and M.U. designed algorithms; A.T. performed simulations.

Competing Interests

The authors declare no competing interests.

Corresponding author

Correspondence to Mihai Udrescu.


text:
# Multiples

Here you will learn about multiples, including how to calculate multiples of a number and solve problems using knowledge of multiples.

Students will first learn about multiples as part of operations and algebraic thinking in elementary school.

## What are multiples?

Multiples are the products you get when two numbers, or factors, are multiplied.

For example, the first five multiples of 7 are 7 , 14 , 21 , 28 , and 35 .

You can skip count on a number line to find multiples of 7 .

The number line shows the first 7 multiples of 7 , but you could keep skipping counting to find more multiples.

To make them easier to find, multiples of the numbers 1 to 9 have patterns. They can help you decide if a given number is a multiple of 1 to 9 .

## Common Core Standards

How does this relate to 4th grade math?

• 4th Grade: Operations and Algebraic Thinking (4.OA.B.4)
Determine whether a given whole number in the range 1 to 100 is a multiple of a given one-digit number.

## How to identify multiples

In order to find if a number is a multiple of another number:

1. Think about the pattern for the multiples of the number.
2. Decide if the number is a multiple.

## Multiples examples

### Example 1: decide whether a number is a multiple

Is 76 a multiple of 6 ?

1. Think about the pattern for the multiples of the number.

Start a list of multiples for 6 :

6 , 12 , 18 , 24 , 30

Notice that the multiples of 6 are also multiples of 2 and 3 .

This pattern is true for all multiples of 6 .

2Decide if the number is a multiple.

✔ The last digit (6) is even, so it is a multiple of 2 .

✘ The sum of the digits (7 + 6 = 13) , so it is not a multiple of 3 .

76 is NOT a multiple of 6 because it is NOT a multiple of 3 .

Skip counting on a number line shows that 76 is NOT a multiple of 6 .

### Example 2: decide whether a number is a multiple

Is 65 a multiple of 5 ?

Start a list of multiples for 5 :

5 , 10 , 15 , 20 , 25

Notice that they end in either a 5 or a 0 .

This pattern is true for all multiples of 5 .

65 is a multiple of 5 because it ends in a 5 .

Skip counting on a number line shows that 65 is the 13^{th} multiple of 5 .

### Example 3: decide whether a number is a multiple

Is 66 a multiple of 4 ?

Start a list of multiples for 4 :

4 , 8 , 12 , 16 , 20

Notice that they are all doubles of a multiple of 2 .

This pattern is true for all multiples of 4 .

66 is the double of 33 \rightarrow 33 \times 2=66 , but 33 is not a multiple of 2 .

66 is not a multiple of 4 because it is not a double of a multiple of 2 .

Skip counting on a number line shows that 66 is NOT a multiple of 4 .

### Example 4: decide whether a number is a multiple

Is 72 a multiple of 9 ?

Start a list of multiples for 9 :

9 , 18 , 27 , 36 , 45

Notice that the sum of the digits is divisible by 9 .

This pattern is true for all multiples of 9 .

72 is a multiple of 9 , because the sum of the digits (7+2=9) is divisible by 9 .

Skip counting on a number line shows that 72 is the eighth multiple of 9 .

### Example 5: decide whether a number is a multiple

Is 42 a multiple of 3 ?

Start a list of multiples for 3 :

3 , 6 , 9 , 12 , 15

Notice that the sum of the digits is divisible by 3 .

This pattern is true for all multiples of 3 .

42 is a multiple of 3 , because the sum of the digits (4+2=6) is divisible by 3 .

Skip counting on a number line shows that 42 is the 14^{th} multiple of 3 .

### Example 6: decide whether a number is a multiple

An elementary school is performing a play. They want to put out 88 chairs for the audience. If they want the same amount of chairs in each row, should they do rows of 6 , 8 or 10 ?

Multiples of 6 : the sum of the digits is divisible by three.

Multiples of 8 : the number is double a multiple of 4 .

Multiples of 10 : the last digit is zero.

88 is NOT a multiple of 6 , because the sum of the digits (8 + 8 = 16) is NOT divisible by 3 .

88 is a multiple of 8 , because it is double a multiple of 4 \rightarrow 2 \times 44=88 .

The school should put the 88 chairs in rows of 8 .

### Teaching tips for multiples

• Use a multiplication table so that students can see examples of multiples and look for patterns.

• While students will need to remember the multiple patterns to solve quickly, avoid just using math worksheets. Help students deepen their understanding and likelihood of remembering by also doing hands-on activities. For example, give students 52 counters and ask them to make as many arrays as possible to find out what factors 52 is a multiple of.

• Play games regularly to help students memorize larger multiples. For example, have students count starting from 1 (the first student says 1 , the second says 2 , the third says 3 and so on), but challenge them to use a different word for a certain multiple each time. So if they were playing with multiples of 7 , whenever a student got to a multiple of 7 , they would say “cookie” instead of the multiple.

### Easy mistakes to make

• Confusing the terms multiples and factors
Factors and multiples are easily mixed up. Remember multiples are the numbers in the middle of the multiplication table, whereas factors are the numbers around the outside (the numbers being multiplied).

• Forgetting the number itself as a multiple
All numbers are a multiple of themselves.
For example, the multiples of 6 are: 6 , 12 , 18 , 24 and so on and so 6 is a multiple of itself.

### Practice multiples questions

1. Which number is a multiple of 2 ?

54

89

77

91

For multiples of 2 , the last digit is an even number. 54 ends in an even number and 27 \times 2=54 , so 54 is a multiple of 2 .

2. Which number is a multiple of 7 ?

99

83

91

74

Using a number line to skip count by 7 , shows that 91 is a multiple of 7 .

3. Which number is NOT a multiple of 3 ?

66

43

39

51

For multiples of 3 , the sum of the digits is divisble by three. 4+3=7 is not divisible by 3 , so 43 is NOT a multiple of 3 .

4. Which number is NOT a multiple of 4 ?

88

68

28

58

For multiples of 4 , the number is double a multiple of 2 . 58 is the double of 29 , so 58 is NOT a multiple of 4 .

5. A grocery store has 78 boxes of cereal. They want to put the same number of boxes on each shelf. How many boxes should they put on each shelf?

4 boxes

5 boxes

6 boxes

7 boxes

For multiples of 6 , the number is divisible by both two and three. 78 is a multiple of 2 , because it ends in an even number (8) .
78 is a multiple of 3 , because 7 + 8 = 15 and 15 \div 3 = 5 .

So 78 is a multiple of 6 \rightarrow 6 \times 13=78
The grocery store can put 6 boxes on each shelf.

## Multiples FAQs

What is the difference between multiples and times tables?

Times tables are a list of multiplication equations for a certain number and include both factors and multiples. Multiples are the answers (products) in times tables.

Can I use division to find multiples?

Yes, if a number divides another number without a remainder, it’s a multiple. For example, in the question “is 72 a multiple of 2 ?” you can solve 72 \div 2=36 . There are no remainders, so it is a multiple.

Do negative numbers have multiples?

All positive and negative numbers have multiples. However, in 4th grade math you only work with positive numbers.

Do prime numbers have multiples?

Prime numbers have an infinite amount of multiples but only 2 factors.

## The next lessons are

• Prime factors
• Factor trees
• Least common multiples (LCM)
• Greatest common factor (GCF)
• Fractions

## Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

format: plain


text:
# SMILES vs. Graph representation in deep learning

I have been reading papers on machine learning and deep learning methods for learning molecular space and generating molecules. These methods use different representations of the molecules. The most popular ones in the field include SMILES and graphs [e.g. this and this]. I have seen a shift of interest from SMILES representations to Graph molecular representations in the past couple years. I was wondering what are possible benefits of graph representations over SMILES?

I can think of two reasons:

1. The uniqueness issue with SMILES: Two SMILE strings might correspond to the same molecule.
2. SMILES are abstract representations while graphs are more natural representations (although, I can't really see this. I mean, I see what it means, but doesn't SMILES also contain the same information?)

EDIT:

some paragraph that I found online and I need interpretation for:

1. it is focused on molecules whose bonds fit the 2-electron valence model
2. it handles a limited array of stereochemistry types
3. there is no standard for handling aromaticity
4. there is no standard way to generate a canonical representation.

are any of these corrected in graphs? From what I know:

For 2: In graphs, it takes more than just defining atoms and bonds to account for stereochemistry. For example, I should also define chirality for each node.

For 3: While in some graph representations, aromatic bonds have their specific type, I found it better to explicitly use the order of the bonds.

For 1 and 4 I don't know if graphs are any better.

• It looks like the 2nd paper about graphs makes some arguments about why they are better (SMILES originate from graph representation anyway, aren't robust to small errors). Feb 21, 2020 at 2:31
• Thanks, I missed that. But their arguments are not that strong. For example, at the end of the day, you need to use some sort of chem package, like RDKit, and that's when you will eventually turn graphs into smiles and eventually mol objects, plus that IMHO the overhead is really negligible. Feb 21, 2020 at 13:59
• Also, I think when the output is categorical, it won't be robust to small perturbations anyways, i.e. small changes in bond and node classes make output graphs invalid. Think of CO2 graph, now shift every class by one (i.e. instead of selecting C from {C, O, N, F, null} select O and instead of selecting double bond from {single, double, triple, null} select triple) it will be ON2 where O and N have triple bonds (and also a single bond between 2 Ns!). Feb 21, 2020 at 14:01
• Please add a citation to the quoted paragraph. If you find a way to canonicalise your representations, then you can fix the problem with aromaticity, too. (It it could be conceptualised accurately in the first place.) You might even be able to fix the stereo chemistry. (That shouldn't be too hard, but it needs to be consistent.) I'm pretty pessimistic about going away from the two electron bond. In the end a string or a graph contain too little information, so you just have to choose which one fits your needs best. Feb 21, 2020 at 20:07

I agree that there seems to be a shift towards using graph representations over SMILES strings. I personally think this is a good thing, and I'll try to explain why, but even if there's nothing inherently better about graph representations of molecules, there is a very practical reason people are moving towards graph representations.

So, first of all, as you point out, both representations suffer from an issue in that they are necessarily a reduction from the complete amount of information needed to specify a molecular structure. For instance, neither a graph nor a SMILES string could distinguish between two isomers which differ only in the direction in which a free $$\ce{O-H}$$ group points. In fact, a bare graph cannot tell the difference between the boat, chair, or planar hexane molecules as they all have identical connectivity. Nonetheless, a graph is a much more flexible representation than a SMILES string because it is quite common to add weights to edges, which could be bond distances, or add parameters to nodes, which could describe angles to other nodes. So, you can include some information which is in the Cartesian coordinates, without having the problems associated with deriving your molecular representation from Cartesian coordinates.

This hints at the very practical reason why people are using graph representations. Basically, graphs are used in machine learning models in many fields other than chemistry. So, it's very nice to not have to reinvent the wheel, especially when people have so much success with things like transfer learning, where you just take a pre-trained model and re-train it for your own purposes.

Also, very often the first step in training a neural network is making some transformation to your data. For instance, graph convolutional neural networks have been successful in many tasks, so why not just use a convolutional filter on your graph representation of a molecule? You could do this with a SMILES string, but you would probably first just transform the string into something resembling a graph.

As to your specific points about chirality and aromaticity, etc. All of this information can be attached to a graph via parameters belonging to each node, although I would personally avoid giving information that isn't strictly necessary. That is, there is nothing special about a bond which is in an aromatic ring. You need to provide enough data that the model can learn about this on its own. If you tell it that bonds in aromatic rings are special enough to get another parameter, this is probably going to bias the model in some unforeseen way. Chirality is easily handled by a simple parameter attached to each node.

Ultimately, though, the representation depends a lot on the problem you are trying to solve. For instance, if you are trying to learn a representation of the potential energy surface, then graphs can work quite well. What is more common perhaps, is to use so-called atom-centered symmetry functions. In this case, the actual features are abstract vectors which are guaranteed to have the relevant symmetries and smoothness needed in the potential energy surface.

If you're doing something more like a classification problem, then using a representation like a SMILES string might be perfectly suitable.

TL;DR

Graphs are a more flexible representation which are commonly used in fields outside of chemistry. Hence, being able to draw on the knowledge of other disciplines is a huge plus, especially when you're collaborating with computer scientists who know a lot about machine learning and nothing about chemistry.

• You mentioned that "neither a graph nor a SMILES string could distinguish between two isomers . . .". Is this also true for the conformers of a single molecule? Nov 27, 2020 at 15:38
• @Blade Yes. The example I gave is a conformer. Both can distinguish between isomers where the actual connectivity is different. For instance, the boat and chair isomers of cyclohexane have the same graphical representation. Nov 29, 2020 at 6:56

It depends on how you code your molecular graphs

The idea of a 'connection table' or valence model for molecules, and thus molecular graphs is embedded in chemical thinking.

1. It's possible to design connection tables that supports a variety of interactions. For example, zero order bonds can encode coordination bonds, delocalized metal-ligand interactions, etc.
2. You can write a graph that stores a variety of stereochemistry, although admittedly axial chirality, etc. require some work to do so (i.e., it's a property of the molecule itself and not any particular atom or bond). Some formats even support concepts like '55% R, 45% S' stereo centers.
3. You would need to define an aromaticity model, although many exist and can be adopted (e.g., 'we use the SMILES aromaticity definition')
4. For both graphs and SMILES, there are many published canonicalization algorithms (e.g., we use the InChI canonical atom order).

In short, people have worried about cheminformatics issues for a long time:

Both papers indicate expansions of the standard molecular graph concepts, e.g. [Gasteiger]:

This representation overcomes the limitations of connection tables designed to only represent chemical structures with bonds localized between two atoms. The representation introduced is based on the separation of the σ- and π-electrons of bonds and the delocalization of electrons also across more than two atoms. It also allows the description of chemical compounds containing multicenter or coordinative bonds.

Alex Clark's zero order bond model linked above gets at many of these issues but in a way that's backwards compatible with the standard SD file format.

It's a long answer, but if you code a good graph representation, you can probably encode a lot of chemistry.


text: 標題: 在某些Cellular Neural Networks的空間與時間的混沌Spatial and temporal chaos in some Cellular Neural Networks 作者: 楊定揮Ting-Hui Yang林松山Song-Sun Lin應用數學系所 關鍵字: 細胞類神經網路;空間混沌;時間混沌;空間熵;Cellular Neural Netwoks;CNN;spatial chaos;temporal chaos;spatial entropy 公開日期: 2002 摘要: 這篇論文主要探討細胞神經網路(CNN)的空間與時間的混沌現象This dissertation investigates the spatial and temporal chaos of some classes of Cellular Neural Networks(CNN). We describe more details as follows. Chapter 1 study the complexity of one-dimensional CNN mosaic patterns with spatially variant templates on finite and infinite lattices. Various boundary conditions are considered for finite lattices and the exact number of mosaic patterns is computed precisely. The entropy of mosaic patterns with periodic templates can also be calculated for infinite lattices. Furthermore, we show the abundance of mosaic patterns with respect to template periods and, which differ greatly from cases with spatially invariant templates. Chapter 2 investigates bifurcations and chaos in two-cells CNN with periodic inputs. Without the inputs, the time periodic solutions are obtained for template $A=[r,p,s]$ with $p>1$, $r>p-1$ and $-s>p-1$. The number of periodic solutions can be proven to be no more than two in exterior region. The input is $b\sin 2\pi t/T$ with period $T>0$ and amplitude $b>0$. The typical trajectories $\Gamma(b,T,A)$ and their $\omega$-limit set $\omega(b,T,A)$ vary with $b$, $T$ and $A$ are considered. The asymptotic limit cycles $\Lambda_\infty(T,A)$ with period $T$ of $\Gamma(b,T,A)$ are obtained as $b\rightarrow\infty$. When $T_0\leq T_0^*$(given in \ref{t0start} ), $\Lambda_\infty$ and $-\Lambda_\infty$ can be separated. The onset of chaos can be induced by crises of $\omega(b,T,A)$ and $-\omega(b,T,A)$ for suitable $T$ and $b$. The ratio $\mathcal{A}(b)=|a_T(b)|/|a_1(b)|$, of largest amplitude $a_1(b)$ and amplitude of the $T$-mode of the Fast Fourier Transform (FFT) of $\Gamma(b,T,A)$, can be used to compare the strength of sustained periodic cycle $\Lambda_0(A)$ and the inputs. When $\mathcal{A}(b)\ll 1$, $\Lambda_0(A)$ dominates and the attractor $\omega(b,T,A)$ is either a quasi-periodic or a periodic. Moreover, the range $b$ of the window of periodic cycles constitutes a devil's staircase. When $\mathcal{A}(b)\sim 1$, finitely many chaotic regions and window regions exist and interweave with each other. In each window, the basic periodic cycle can be identified. A sequence of period-doubling is observed to the left of the basic periodic cycle and a quasi-periodic region is observed to the right of it. For large $b$, the input dominates, $\omega(b,T,A)$ becomes simpler, from quasi-periodic to periodic as $b$ increases. URI: http://140.113.39.130/cdrfb3/record/nctu/#NT910507027http://hdl.handle.net/11536/70960 Appears in Collections: Thesis


text:
# Photon Energy Problem

What is the energy carried by a mole of photons of wavelength 522nm which is associated with green light?

$$E=h \nu$$
Where h is Planck's constant and $$\nu$$ is the frequency. So you have to express the frequency through the wavelength.


text:
# Laboratoire de Physique ThéoriqueToulouse - UMR 5152

## Rechercher

Sur ce site

Sur le Web du CNRS

Accueil du site > Séminaires > Séminaires 2015 > Many Body Localization Transition : level statistics and entanglement entropy

Mardi 10 novembre 2015-14:00

## Many Body Localization Transition : level statistics and entanglement entropy

Cécile Monthus (IPhT, CEA Saclay-CNRS)

par - 10 novembre 2015

At the Many-Body-Localization (MBL) Transition, the statistics of energy levels and the entanglement entropy of eigenstates change : the Many-Body-Localized phase is characterized by the Poisson level statistics and an area-law for the entanglement, while the Delocalized phase is characterized by the Wigner-Dyson level statistics and a volume-law for the entanglement. In the first part of the talk, the Dyson Brownian Motion approach for quantum spin Hamiltonians with random fields will be described : the statistics of energy levels can be studied via Langevin and Fokker-Planck equations, and one obtains the level repulsion exponent $\beta$ in terms of the Edwards-Anderson matrix elements. In the second part of the talk, we will consider the strong disorder limit of the MBL transition, where the critical level statistics is close to the Poisson statistics, in order to determine the statistical properties of the rare extensive resonances that are needed to escape from the area-law entanglement of the Localized phase. At criticality, the entanglement entropy can grow with an exponent $0<\alpha < 1$ anywhere between the area law $\alpha=0$ and the volume law $\alpha=1$, as a function of the resonances properties. In addition, the correlation length exponent takes the simple value $\nu=1$. Independently of this strong disorder limit, we will explain why for the Many-Body-Localization transition concerning individual eigenstates, the correlation length exponent $\nu$ is not constrained by the usual Harris inequality $\nu \geq 2/d$.

Post-scriptum :

contact : F. Alet / N. Laflorencie


text:
# The Giant Radio Array for Neutrino Detection (GRAND) Project

### Submission summary

 As Contributors: Bruno Lazarotto Lago Arxiv Link: https://arxiv.org/abs/2110.14417v2 (pdf) Date accepted: 2022-03-22 Date submitted: 2022-02-04 14:10 Submitted by: Lazarotto Lago, Bruno Submitted to: SciPost Physics Proceedings Proceedings issue: 50th International Symposium on Multiparticle Dynamics (ISMD2021) Academic field: Physics Specialties: High-Energy Physics - Experiment

### Abstract

GRAND is designed to detect ultra-high-energy cosmic particles -- specially neutrinos, cosmic rays and gamma rays using radio antennas. With $\sim$20 mountainous sites around the world it will cover a total area of 200,000 km$^{2}$. The planned sensitivity of 10$^{-10}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$ above $5\times10^{17}$ eV will likely ensure the detection of cosmogenic neutrinos predicted by most common scenarios enabling neutrino astronomy. Furthermore, PeV--EeV neutrinos can test particle interactions at energies above those achieved in accelerators. The pathfinder stage GRANDProto300 is planned to start taking data in 2021. We present the current overall status of the project with emphasis on the neutrino physics.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics Proceedings: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

I would like to thank the referee for the feedback, suggestions and comments I received.
Alongside the changes requested regarding grammar, style and citations, in section 2, I made the increase in the exposure clearer mentioning it is to be compared with the exposure from the Pierre Auger Observatory. I also added a sentence on section 4.3 addressing the muon discrepancy mentioned earlier on the text.

### List of changes

- Section 1, paragraph 2, Ref [2]: Included a citation to https://arxiv.org/abs/1903.07713
- Section 2, paragraph 1: I made the increase in the exposure clearer mentioning it is to be compared with the exposure from the Pierre Auger Observatory
- Section 2, paragraph 3: "involved on the" -> "involved in the"
- Section 2, paragraph 4: Made it clear that this is the sensitivity for gamma rays.
- Section 3, paragraph 3: "in the atmosphere and also" removed "and"
- Section 4.1, paragraph 1: Removed "(EBL)"
- Section 4.1, last paragraphs: Merged the last three paragraphs.
- Section 4.2, last bullet: Removed extra space at "(CRB) ." and "to" in "(up to to 10Mpc)".
- Section 4.3: I added a sentence addressing the muon discrepancy mentioned earlier on the text.
- Section 4.4, paragraph 2: Put a comma between "timing making".
- Section 4.4, paragraph 3: Changed "detected by Laser ... Observatories - LIGO..." to "detected by the Laser ... Observatories LIGO..."


text:
## continuous herded Gibbs sampling

Posted in Books, pictures, Statistics with tags , , , , , , , , on June 28, 2021 by xi'an

Read a short paper by Laura Wolf and Marcus Baum on Gibbs herding, where herding is a technique of “deterministic sampling”, for instance selecting points over the support of the distribution by matching exact and empirical (or “empirical”!) moments. Which reminds me of the principal points devised by my late friend Bernhard Flury. With an unclear argument as to why it could take over random sampling:

“random numbers are often generated by pseudo-random number generators, hence are not truly random”

Especially since the aim is to “draw samples from continuous multivariate probability densities.” The sequential construction of such a sample proceeds sequentially by adding a new (T+1)-th point to the existing sample of y’s by maximising in x the discrepancy

$(T+1)\mathbb E^Y[k(x,Y)]-\sum_{t=1}^T k(x,y_t)$

where k(·,·) is a kernel, e.g. a Gaussian density. Hence a complexity that grows as O(T). The current paper suggests using Gibbs “sampling” to update one component of x at a time. Using the conditional version of the above discrepancy. Making the complexity grow as O(dT) in d dimensions.

I remain puzzled by the whole thing as these samples cannot be used as regular random or quasi-random samples. And in particular do not produce unbiased estimators of anything. Obviously. The production of such samples being furthermore computationally costly it is also unclear to me that they could even be used for quick & dirty approximations of a target sample.


text:
Bohnsack for assist with oocyte dissections, and Kerstin Hupfeld for excellent techie assistance

Bohnsack for assist with oocyte dissections, and Kerstin Hupfeld for excellent techie assistance. put in place specialized complexes known as spliceosomes, where factors necessary for splicing of pre-mRNAs are enriched. Presently, 150 different protein and several little RNAs have already been identified as component of spliceosomes, that are arranged in specific subcomplexes. One of the most prominent spliceosome subunits will be the uridine-rich little nuclear RNPs (U snRNPs) from the Sm course. They contain an RNA element (uridine-rich little nuclear RNA [U snRNA]) and many protein that are either common for everyone or specific for just one particle (for review discover Nilsen, 2003). Though splicing takes place in the nucleus Also, major elements of the biogenesis of U snRNPs happen in the cytoplasm. The nuclear-encoded m7G-capped U snRNA is certainly transiently exported towards the cytoplasm to permit binding of the normal (Sm) proteins. This qualified prospects to the forming of the Sm primary area, the structural construction of most spliceosomal U snRNPs from the Sm course (Raker et al., 1996). Development from the Sm primary is necessary for cover hypermethylation and the next nuclear import of U snRNPs (Hamm et al., 1990). Inside the nucleus, U snRNPs are LRRC63 initial geared to subnuclear domains termed Cajal physiques (CBs), where extra modifications in the RNA take place with least some particular protein are added. Ultimately, the older U snRNPs migrate to perichromatin fibrils, the websites of transcription and splicing (for testimonials discover Meister et al., 2002; Matera et al., 2007). Oddly enough, recent research indicated that many areas of the biogenesis routine of U snRNPs are AM 103 aspect mediated and governed in vivo. One of the most prominent AM 103 element in this process may be the success electric motor neuron (SMN) complicated, a macromolecular entity that positively mediates the binding of the normal Sm protein onto U snRNAs. This complicated includes nine main proteins, like the SMN gene item, Gemin2C8, as well as the unr-interacting proteins (unrip; for review articles discover Meister et al., 2002; Gubitz et al., 2004; Pellizzoni, 2007) (Carissimi et al., 2005; Grimmler et al., 2005b). Another complicated handles The SMN complicated, whose name-giving component may be the type-II proteins arginine methyltransferase 5. This device, together with various other elements perhaps, changes arginine residues in a few Sm proteins into symmetrical dimethylarginines, thus improving their affinity for the SMN complicated and stimulating U snRNP set up (Brahms et al., 2001; Friesen et al., 2001; Meister et al., 2001b; Fischer and Meister, 2002). Furthermore, it’s been shown the fact that SMN complicated (or parts thereof) also take part in the next nuclear import of U snRNPs (Narayanan et al., 2004; Matera and Shpargel, 2005). Once in the nucleus, both products migrate to CBs, where in fact the SMN complicated accumulates and U snRNPs are released to sites of splicing after extra maturation guidelines (Stanek and Neugebauer, 2006). These observations claim that U snRNPs dissociate from SMN complexes in CBs which the SMN complicated returns as another unit towards the cytoplasm at afterwards stages. Even though the cytoplasmic role from the SMN complicated is certainly understood in a few detail, its features in the nucleus are just characterized poorly. Thus, it really is still unclear how U snRNPs are separated through the SMN complicated after nuclear import and the way the return from the SMN complicated towards the cytoplasm is certainly facilitated. A significant participant in this technique could be unrip, which interacts using the SMN complicated in the AM 103 cytoplasm primarily. Knockdown of the factor qualified prospects to enhanced deposition.


text:
## Theories for incompressible rods: a rigorous derivation via $$\Gamma$$-convergence.(English)Zbl 07446485

Summary: We use variational convergence to derive a hierarchy of one-dimensional rod theories, starting out from three-dimensional models in nonlinear elasticity subject to local volume-preservation. The densities of the resulting $$\Gamma$$-limits are determined by minimization problems with a trace constraint that arises from the linearization of the determinant condition of incompressibility. While the proofs of the lower bounds rely on suitable constraint regularization, the upper bounds require a careful, explicit construction of locally volume-preserving recovery sequences. After decoupling the cross-section variables with the help of divergence-free extensions, we apply an inner perturbation argument to enforce the desired non-convex determinant constraint. To illustrate our findings, we discuss the special case of isotropic materials.

### MSC:

 35Qxx Partial differential equations of mathematical physics and other areas of application
Full Text:

### References:

 [2] E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string,J. Elasticity 25(2) (1991), 137-148. doi:10.1007/BF00042462. · Zbl 0734.73094 [3] J. Adams, S. Conti and A. DeSimone, Soft elasticity and microstructure in smectic C elastomers,Contin. Mech. Thermodyn.18(6) (2007), 319-334. doi:10.1007/s00161-006-0031-8. · Zbl 1170.76303 [4] M. Amabili, I.D. Breslavsky and J.N. Reddy, Nonlinear higher-order shell theory for incompressible biological hyperelastic materials,Comput. Methods Appl. Mech. Engrg.346(2019), 841-861. doi:10.1016/j.cma.2018.09.023. · Zbl 1440.74063 [5] H. Attouch, Variational convergence for functions and operators,Applicable Mathematics Series. Pitman (Advanced Publishing Program)(1984). · Zbl 0561.49012 [6] S. Bartels, finite element simulation of nonlinear bending models for thin elastic rods and plates, in:Geometric Partial Differential Equations - Part I, A. Bonito and R.H. Nochetto, eds, Handbook of Numerical Analysis, Vol. 21, Elsevier, 2020, pp. 221-273, Chapter 3. doi:10.1016/bs.hna.2019.06.003. · Zbl 1455.35255 [7] K. Bhattacharya, M. Lewicka and M. Schäffner, Plates with incompatible prestrain,Arch. Ration. Mech. Anal.221(1) (2016), 143-181. doi:10.1007/s00205-015-0958-7. · Zbl 1382.74081 [8] A. Braides,-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications., Vol. 22, Oxford University Press, Oxford, 2002. · Zbl 1198.49001 [9] M. Bukal, M. Pawelczyk and I. Velˇci´c, Derivation of homogenized Euler-Lagrange equations for von Kármán rods, J. Differential Equations262(11) (2017), 5565-5605. doi:10.1016/j.jde.2017.02.009. · Zbl 1358.74008 [10] M. Chermisi and H. Li, The von Kármán theory for incompressible elastic shells,Calc. Var. Partial Differential Equations 48(1-2) (2013), 185-209. doi:10.1007/s00526-012-0549-5. · Zbl 1314.74043 [11] P. Ciarlet,Mathematical Elasticity: Theory of Plates. Developments in Aquaculture and Fisheries Science, North-Holland, 1997. [12] M. Cicalese, M. Ruf and F. Solombrino, Hemihelical local minimizers in prestrained elastic bi-strips,Z. Angew. Math. Phys.68(6):Art(2017), 122, 18. · Zbl 1395.74075 [13] M. Cicalese, M. Ruf and F. Solombrino, On global and local minimizers of prestrained thin elastic rods,Calc. Var. Partial Differential Equations56(4):Art(2017), 115, 34. · Zbl 1371.74164 [14] S. Conti and G. Dolzmann, Derivation of elastic theories for thin sheets and the constraint of incompressibility, in:Analysis, Modeling and Simulation of Multiscale Problems, Springer, Berlin, 2006, pp. 225-247. doi:10.1007/3-540-356576_9. · Zbl 1366.74043 [15] S. Conti and G. Dolzmann,-convergence for incompressible elastic plates,Calc. Var. Partial Differential Equations 34(4) (2009), 531-551. doi:10.1007/s00526-008-0194-1. · Zbl 1161.74038 [16] S. Conti and G. Dolzmann, On the theory of relaxation in nonlinear elasticity with constraints on the determinant,Arch. Ration. Mech. Anal.217(2) (2015), 413-437. doi:10.1007/s00205-014-0835-9. · Zbl 1323.49010 [17] G. Dal Maso,An Introduction to-convergence, Progress in Nonlinear Differential Equations and their Applications., Vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. · Zbl 0816.49001 [18] E. Davoli and M.G. Mora, Convergence of equilibria of thin elastic rods under physical growth conditions for the energy density,Proc. Roy. Soc. Edinburgh Sect. A142(3) (2012), 501-524. doi:10.1017/S0308210510001563. · Zbl 1317.74051 [19] A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies,Arch. Ration. Mech. Anal.161(3) (2002), 181-204. doi:10.1007/s002050100174. · Zbl 1017.74049 [20] K.O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn’s inequality,Ann. of Math. (2)48 (1947), 441-471. doi:10.2307/1969180. · Zbl 0029.17002 [21] G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity,Comm. Pure Appl. Math.55(11) (2002), 1461-1506. doi:10.1002/cpa.10048. · Zbl 1021.74024 [22] G. Friesecke, R.D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gammaconvergence,Arch. Ration. Mech. Anal.180(2) (2006), 183-236. doi:10.1007/s00205-005-0400-7. · Zbl 1100.74039 [23] V. Girault and P.-A. Raviart,Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, Vol. 5, Springer-Verlag, Berlin, 1986, Theory and algorithms. · Zbl 0585.65077 [24] P. Hornung, Invertibility and non-invertibility in thin elastic structures,Arch. Ration. Mech. Anal.199(2) (2011), 353-368. doi:10.1007/s00205-010-0391-x. · Zbl 1294.74048 [25] P. Hornung, S. Neukamm and I. Velˇci´c, Derivation of a homogenized nonlinear plate theory from 3d elasticity,Calc. Var. Partial Differential Equations51(3-4) (2014), 677-699. doi:10.1007/s00526-013-0691-8. · Zbl 1327.35018 [26] T. Kato, M. Mitrea, G. Ponce and M. Taylor, Extension and representation of divergence-free vector fields on bounded domains,Math. Res. Lett.7(5-6) (2000), 643-650. doi:10.4310/MRL.2000.v7.n5.a10. · Zbl 0980.53022 [27] R.V. Kohn and E. O’Brien, On the bending and twisting of rods with misfit,J. Elasticity130(1) (2018), 115-143. doi:10. 1007/s10659-017-9635-4. · Zbl 1387.74065 [28] V.A. Kondratev and O.A. Oleinik, Boundary-value problems for the system of elasticity theory in unbounded domains. korn’s inequalities,Russian Mathematical Surveys43(5) (1988), 65-119. doi:10.1070/RM1988v043n05ABEH001945. · Zbl 0669.73005 [29] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9)74(6) (1995), 549-578. · Zbl 0847.73025 [30] H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity: A variational asymptotic derivation,Journal of Nonlinear Science6(1) (1996), 59-84. · Zbl 0844.73045 [31] M. Lewicka and H. Li, Convergence of equilibria for incompressible elastic plates in the von Kármán regime,Commun. Pure Appl. Anal.14(1) (2015), 143-166. doi:10.3934/cpaa.2015.14.143. · Zbl 1317.74058 [32] M.G. Mora and S. Müller, Derivation of the nonlinear bending-torsion theory for inextensible rods by-convergence, Calc. Var. Partial Differential Equations18(3) (2003), 287-305. doi:10.1007/s00526-003-0204-2. · Zbl 1053.74027 [33] M.G. Mora and S. Müller, A nonlinear model for inextensible rods as a low energy-limit of three-dimensional nonlinear elasticity,Ann. Inst. H. Poincaré Anal. Non Linéaire21(3) (2004), 271-293. · Zbl 1109.74028 [34] M.G. Mora and S. Müller, Convergence of equilibria of three-dimensional thin elastic beams,Proc. Roy. Soc. Edinburgh Sect. A138(4) (2008), 873-896. doi:10.1017/S0308210506001120. · Zbl 1142.74022 [35] M.G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density,J. Differential Equations252(1) (2012), 35-55. doi:10.1016/j.jde.2011.09.009. · Zbl 1291.74128 [36] S. Müller and M.R. Pakzad, Convergence of equilibria of thin elastic plates—the von Kármán case,Comm. Partial Differential Equations33(4-6) (2008), 1018-1032. doi:10.1080/03605300701629443. · Zbl 1141.74034 [37] S. Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity,Arch. Ration. Mech. Anal.206(2) (2012), 645-706. doi:10.1007/s00205-012-0539-y. · Zbl 1295.74049 [38] S. Neukamm and I. Velˇci´c, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity,Math. Models Methods Appl. Sci.23(14) (2013), 2701-2748. doi:10.1142/S0218202513500449. · Zbl 1282.35039 [39] R.W. Ogden, Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids,Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences326(1567) (1972), 565-584. · Zbl 0257.73034 [40] H. Olbermann and E. Runa, Interpenetration of matter in plate theories obtained as-limits,ESAIM Control Optim. Calc. Var.23(1) (2017), 119-136. doi:10.1051/cocv/2015042. · Zbl 1364.49015 [41] O. Pantz, On the justification of the nonlinear inextensional plate model,Arch. Ration. Mech. Anal.167(3) (2003), 179- 209. doi:10.1007/s00205-002-0238-1. · Zbl 1030.74031 [42] J. Peypouquet,Convex Optimization in Normed Spaces: Theory, Methods and Examples, Springer Briefs in Optimization, Springer, Cham, 2015, With a foreword by Hedy Attouch. · Zbl 1322.90004 [43] L. Scardia, Asymptotic models for curved rods derived from nonlinear elasticity by-convergence,Proc. Roy. Soc. Edinburgh Sect. A139(5) (2009), 1037-1070. doi:10.1017/S0308210507000194. · Zbl 1422.74061 [44] L. Trabucho and J.M. Viaño, Mathematical modelling of rods, in:Handbook of Numerical Analysis, Vol. IV, Handb. Numer. Anal., Vol. IV, North-Holland, Amsterdam, 1996, pp. 487-974 · Zbl 0873.73041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.