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Abstract

Understanding the structural organization of ingredient relationships within cuisines can reveal fundamental patterns in culinary traditions and ingredient co-occurrence. In this paper, we constructed ingredient networks (InNs) from two datasets of recipe ingredients that include recipes from 10 cuisines around the world. We then performed an empirical investigation of these multicuisine InNs to examine their structural characteristics. Our analysis demonstrates that networks exhibit scale-free behavior, with degree distributions following a power law characterized by exponents ranging from $γ = 1.96$ to $γ = 2.38$ . This further aligns with statistical validation, where the $R$ -squared values range from 0.9965 to 0.9991, and the $p$ -values are extremely low ( $10^{- 25}$ to $10^{- 30}$ ), reinforcing the robustness of the power law fit. Additionally, the networks display ultra-small-world properties, as evidenced by their short network diameter of approximately 4. These structural measurements highlight striking similarities between InNs and widely studied social networks, suggesting underlying patterns reflective of social-like dynamics. Furthermore, the communities formed within these InNs show a strong correlation with the categorical grouping of recipes, providing insights into the evolution of culinary traditions and ingredient compatibility.

Keywords

social network analysis ingredient network power law social network metrics community structure

1. Introduction

A social network can be represented as a graph consisting of individuals or entities, their interconnections, and the modes of communication between them (Ghani et al., 2019; Newman et al., 2002). Social network analysis (SNA) is a method used to examine and study the different aspects and characteristics of these networks (Liu et al., 2019; Tabassum et al., 2018). Although SNA was developed with networks formed by human society, it has now been utilized in many different types of networks. SNA principles may be equally relevant to understanding the structure and dynamics of ingredient networks (InNs), where ingredients and their relationships form the basis of the network.

Unlike random networks, which exhibit a homogeneous distribution of degrees, real-world networks, such as InNs, often follow a scale-free structure. Scale-free networks are characterized by a power-law degree distribution, where a few highly connected nodes (hubs) coexist with many sparsely connected nodes (Barabási & Albert, 1999). This structure arises from growth and preferential attachment, where highly connected nodes are more likely to attract new connections. In the context of InNs, this implies that certain ingredients (hubs) play a central role in connecting various components, while others remain peripheral.

Understanding the scale-free nature of InNs is crucial for analyzing their structure and evolution. For example, ingredients in the central region can represent foundational or widely used components, while sparsely connected ingredients can indicate niche or specialized elements. In addition, the formation of communities within such networks can reveal patterns of ingredient usage and compatibility, offering insight into culinary practices and recipe design. Figure 1 illustrates the distinction between random and scale-free networks, highlighting the presence of hub nodes in the latter. In InNs, these hubs are critical for maintaining connectivity and facilitating interactions between less connected ingredients. Using SNA techniques, this study aims to uncover the underlying structure of InNs, identify key ingredients, and explore their roles in the shaping of culinary systems.

Figure 1.

Random network versus scale-free network.

In this study, we draw a conceptual parallel between InNs and social networks by demonstrating that InNs exhibit structural and organizational properties commonly observed in social systems. While traditional social networks model relationships between individuals or entities, InNs capture the co-occurrence and collaborative relationships between ingredients in recipes. These relationships give rise to patterns such as scale-free degree distributions, community structures, and centrality hierarchies, which are hallmarks of social networks. The emergence of these patterns suggests that InNs, such as social networks, are governed by the principles of preferential attachment, collaboration, and modular organization. By applying social network metrics, such as centrality measures, diameter, and community detection, we not only quantify these properties, but also provide a novel perspective on the organizational dynamics of culinary systems. This analogy strengthens the case for interpreting ingredient interactions as a form of “social” behavior, where ingredients “collaborate” in recipes to create complex flavor profiles, much like individuals collaborate in social systems to achieve collective outcomes.

This paper analyzes the Indian dataset of recipes and ingredients (INDoRI), which encompasses a total of 5,187 recipes, and the Yummly dataset consists of multi-cuisine details spanning more than 12 global cuisines. In addition, the dataset includes a graph-based representation of the relationships between ingredients, that is, the InN. InN is formed by capturing ingredient relationships based on their co-occurrence within recipes. InN is essentially a graph $G (V, E, w)$ of ingredients (Nyati et al., 2021), constructed by joining ingredients that appear together in different recipes. The vertices $V$ represent the ingredients, the edges $E$ contain the connection between the vertices, and the weight $w$ represents the strength of the association. The association is stronger if the ingredients appear multiple times in different recipes. For example, the ingredients “onion” and “tomato” have a strong association as they appear numerous times in many recipes (refer Figure 3). The dataset and its characteristics were previously published in Khanna et al. (2023). Extended research involves studying the properties of the dataset and comparing it with other cuisines through a detailed empirical analysis. This work is emphasized in this paper.

Figure 2.

Key characteristics of Indian dataset of recipes and ingredients (INDoRI).

Figure 3.

Ingredient network (InN) representation.

Any study on a novel network starts with empirical analysis to know whether the network is generated randomly or through preferential attachment. In the same sense, it is interesting to know the characteristics of InN. The heterogeneous degree distribution in InNs becomes evident as certain ingredients are essential in numerous recipes while others are only present in a few dishes. This observation highlights the varying degrees of ingredient usage, emphasizing each ingredient’s diverse role within the network. However, are these differences in degree significant enough to call this network a social network? Does the degree of this network follow a power law distribution? In this paper, we have addressed these questions through an extensive empirical analysis.

The organization of the remainder of this paper is as follows: Section 2 provides an overview of the related work on SNA and InNs. Section 3 explains the methodology, datasets, and metrics involved. Section 4 displays the SNA experimental results. Section 5 offers an in-depth discussion, while Section 6 explores the applications. Finally, Section 7 wraps up with conclusions and suggests future research avenues.

2. Related Work

The theory of complex networks characterizes social networks by a scale-free property (Barabási & Bonabeau, 2003). This property allows it to possess specific properties different from the random network. Although the study of SNA started with the human relational network (Moreno, 1941), researchers have investigated and explored various other networks and found that many are structurally similar to social networks (Wang, 2017). The original attempt of Watts and Strogatz in their work on small-world networks (Watts & Strogatz, 1998) was to construct a network model with a small average path length as a random graph and a relatively large clustering coefficient as a regular lattice, which evolved to become a new network model in its current form. However, the discovery of scale-free networks was based on the observation that the degree distributions of many real networks have a power-law form, albeit with power-law distributions. Another significant recent discovery is that many large-scale complex networks are scale-free; that is, their connectivity distributions are in a power-law form independent of the network scale (Barabási & Albert, 1999; Barabási et al., 1999). Unlike an exponential network, a scale-free network is homogeneous: most nodes have very few link connections, yet a few nodes have many connections.

SNA has been applied to various networks, some of the notable examples being Twitter (Cataldi et al., 2010; Mohammad et al., 2016; Zhang et al., 2015), Facebook (Chandola et al., 2009; Zhang et al., 2015), human interaction networks (Cho et al., 2012), internet (Vazquez et al., 2002), WWW (Albert et al., 1999), email (Ebel et al., 2002), software (Valverde et al., 2002), electronic circuits (Janssen C et al., 2001), language (Cancho & Solé, 2001), movie actors (Barabási & Albert, 1999; Watts & Strogatz, 1998), math coauthorship (Newman, 2001), food web (Montoya & Solé, 2002; Williams et al., 2002), and metabolism (Jeong et al., 2000), where these networks were analyzed and proved that they all follow the small world pattern and are scale-free. In contrast, analysis of the biomedical research collaboration network (Bian et al., 2014) shows that collaboration networks are not scale-free but have small-world properties. Similarly, the power grid network is also not scale-free (Pagani & Aiello, 2013).

InNs have been widely studied to understand the relationships between food components, culinary traditions, and cultural influences. Researchers have investigated InNs within different contexts, including identification of communities (Teng et al., 2012), recipe recommendation (Nyati et al., 2021), and recipe recognition (Min et al., 2019). Like Herrera (2021), offers a comprehensive review of network science applications in food studies, which contextualizes our work within the broader field. Similarly, studies by Ahn et al. (2011) and Ahnert (2013) provide insight into flavor networks and food pairing principles, which complement our analysis of ingredient connectivity and co-occurrence patterns. Based on this, Teng et al. (2012) developed an InN-based recipe recommendation system, showing that ingredient connectivity can be used to customize recipe suggestions. Their approach highlighted the role of network structures in food preferences, yet it primarily focused on algorithmic recommendations rather than structural properties of multi-cuisine networks. More recently, Shirai et al. (2021) used semantic knowledge graphs to model ingredient relationships, allowing nutritional reasoning and nutrition-based food suggestions. Their work integrated ontology-based analysis, while our study focuses on empirical network structures in real-world recipe datasets. Furthermore, Cheng et al. (2021) explored InNs from a health perspective, applying natural language processing (NLP) and content analysis to assess the nutritional quality of recipes shared online. Although their study focused on dietary health, our research examines InNs from a structural and cultural perspective, revealing insights into how culinary traditions shape ingredient co-occurrence patterns. These prior works establish the significance of InN analysis, and our study extends this research by providing a multi-cuisine comparative analysis with a focus on Indian and international culinary structures.

By integrating these references, our study is positioned as an extension and refinement of previous research, moving beyond flavor-based networks to a more structural and community-based understanding of InNs across global cuisines. Unlike previous works that focus primarily on ingredient similarity and pairing principles, our study introduces a deeper network-based analysis that incorporates metrics such as eigen centrality (EC), clustering coefficients, and modularity to capture the hierarchical structure of the usage of culinary ingredients. Furthermore, by analyzing 10 global cuisines, we provide a comparative perspective that broadens the scope of existing research, offering novel insights into the social-like structures and small-world properties of InNs.

Thus, while the existing literature lays a strong foundation for food network analysis, our work contributes by introducing a more comprehensive, structural, and empirical analysis of InNs across multiple cuisines, offering new perspectives on how culinary traditions evolve through network-based interactions. However, none of these studies investigates whether InN shows randomness or scale-free characteristics.

3. Methodology

The overall pipeline is structured into several key stages: initial introduction of the dataset, followed by cleanup and the construction of an InN. Subsequently, the focus is on analyzing various cuisines using social network metrics. In addition, the investigation explores the communities within InN, comparing these findings across different community detection algorithms. These steps are elaborated in the subsequent sections.

3.1. Indian dataset of recipes and ingredients (INDoRI)

Creating a comprehensive dataset of Indian cuisines poses unique challenges. One of them is to compile recipes that span the diverse cultural landscape of India. Due to the same reason, one may not find all the recipes from one single web portal. As there is no common data format available, each portal presents data differently, and the data is unstructured. Hence, the second challenge is to extract meaningful information from it. We consider multiple recipe websites to address the first challenge. All the unstructured data therein are crawled using a Python script.

The distinctive feature of INDoRI compared to other food datasets is its collection of 5,187 recipes spanning 18 unique Indian cuisines. In addition, it includes additional details such as cuisine, category, cooking time, and cooking instructions. The recipes are organized into eight specific categories. Excluding the 925 uncategorized recipes, the remaining are also divided among 18 distinct cuisines. Table 1 provides a comparison of INDoRI with existing datasets related to Indian food. Figure 2 highlights the primary characteristics of INDoRI.

Table 1.
Comparison of Indian Dataset of Recipes and Ingredients (INDoRI) With Other Datasets.

Dataset name No. of recipes Ingredients Category} Type Cooking instructions Time to prepare Regional information Image link Recipe link Corresponding graphical data

Recipe ingredients dataset (Yummly, 2024) 12,000 $+$ Yes Yes No No No Yes No No No

Indian food dataset (Jain, 2020) 6,000 Yes No Yes Yes Yes Yes No Yes No

Indian Food 101 (Prabhavalkar, n.d) 255 Yes No Yes No Yes Yes No No No

INDoRI (Khanna et al., 2023) 5,187 Yes Yes Yes Yes Yes Yes Yes Yes Yes

Dataset name	No. of recipes	Ingredients	Category}	Type	Cooking instructions	Time to prepare	Regional information	Image link	Recipe link	Corresponding graphical data
Recipe ingredients dataset (Yummly, 2024)	12,000 $+$	Yes	Yes	No	No	No	Yes	No	No	No
Indian food dataset (Jain, 2020)	6,000	Yes	No	Yes	Yes	Yes	Yes	No	Yes	No
Indian Food 101 (Prabhavalkar, n.d)	255	Yes	No	Yes	No	Yes	Yes	No	No	No
INDoRI (Khanna et al., 2023)	5,187	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes

Detailed information on the INDoRI dataset, the cleaning process, and ingredient stop words (ISWs) can be found in Khanna et al. (2023).

3.2. Dataset for Empirical Analysis

We used two datasets to create, analyze, and understand InNs. The first dataset, INDoRI, is a comprehensive compilation of Indian recipes and their corresponding ingredients. In addition to the INDoRI dataset, we incorporated the recipe ingredient dataset sourced from Yummly (2024). The primary reason for using the Yummly dataset was to allow for a comparative analysis of the properties of ingredients in diverse global cuisines. Although INDoRI provides a rich and unique representation of Indian cuisine, the inclusion of Yummly’s dataset allows us to contextualize our findings within a broader global framework. This comparative approach is essential for understanding how the usage patterns of ingredients in Indian cuisine differ from or align with those of other cuisines.

The Yummly dataset contains ingredient information from more than 14 global cuisines, and we deliberately selected and filtered nine cuisines based on their popularity and the quantity of available records. Combined with the Indian cuisine data from INDoRI, our study focuses on analyzing InNs of a total of 10 cuisines. Importantly, the INDoRI dataset remains central to our analysis, as it provides a detailed and culturally specific representation of Indian cuisine, which is not available in the Yummly dataset. By performing combined and separate analyses, we ensure that the unique characteristics and contributions of INDoRI are highlighted, while also leveraging the Yummly dataset to provide a global perspective.

The ingredients contained extraneous details beyond the ingredient names, requiring the removal of specific words such as “cup,” “chopped,” and “boiled,” categorized as ISWs. This filtering process, based on Khanna et al. (2023), refines the ingredient names for a cleaner dataset. For example, the phrase “half-cup of sugar” is reduced to “sugar” through a sequence of actions. Initially, ISWs are eliminated, leading to the removal of the words “half” and “cup.” The word “of” is also discarded as an NLP stop word, thus facilitating a more precise InN. The detailed process is provided in the reference source (Khanna et al., 2023).

3.3. InN Construction

We developed an InN, represented as a graph $G (V, E, w)$ , where $V$ is a set of ingredients, $E$ is the connection between ingredients and $w : V \times V \to R$ of an edge signifies the number of associations between ingredients in different recipes. Here, each node corresponds to an ingredient, and an edge is formed between two nodes if the corresponding ingredients appear together in the same recipe. The more they appear together in diverse recipes, the stronger the association. A visual representation of this network for Indian cuisine is illustrated in Figure 3, where the thickness of an edge correlates with the strength of the association. In particular, the strongest link is between salt and oil, which co-occur in 1,523 recipes, highlighted by one of the thickest edges in the graph.

3.4. Social Metrics Evaluation

In order to examine the dynamics and behavior of InN, we utilized a range of social network measurements, which we classified into macro and micro metrics. Macro metrics, such as distance, diameter, and density, provide valuable information about the overall structure and connectivity of the network. However, micro-metrics, such as node degree, clustering coefficient, closeness centrality (CC), and eigencentrality, offer a detailed perspective by assessing the behavior and roles of individual components in the network. Specifically, CC helps identify ingredients that are closely connected to others, suggesting their potential as central or bridging components in the network. Eigencentrality, on the other hand, highlights ingredients that are connected to other highly connected ingredients, indicating their influence and importance in the network. These metrics collectively provide information on the functional roles and relative significance of individual ingredients, allowing a deeper understanding of their usage patterns and interactions within the network. The results obtained from these metrics are explained in Section 4, demonstrating how each statistic adds to a thorough understanding of the InN features.

3.5. Community Structure Identification

Scale-free networks exhibit a community structure. In the course of studying and analyzing InN, we also tried to identify communities within InN. We used various community detection algorithms to partition the InN into multiple communities. Specifically, we implemented weighted versions of the Leiden (Traag et al., 2019), Louvain (Blondel et al., 2008), and weighted association-based community detection (WABCD; Khanna et al., 2023) algorithms. It was observed that InN also exhibits a community structure such as other scale-free networks. In particular, the weighted Louvain and Leiden algorithms organized communities based on a weighted modularity score, while the WABCD method grouped ingredients by the strength of their connections. We used standard implementations of the Leiden, Louvain, and WABCD algorithms for weighted community detection, ensuring edge weights represented ingredient co-occurrence frequencies. The resolution parameter was set to 1.0 for both Leiden and Louvain to maintain a consistent granularity of the community. WABCD followed weight-based adjustments as per Vazquez et al. (2002). These algorithms were chosen for their effectiveness: Leiden optimizes modularity for well-separated communities, Louvain provides fast hierarchical clustering, and WABCD is tailored for weighted networks, making it ideal for ingredient co-occurrence analysis.

The details of these community structures and their implications are further discussed in Section 4.

4. Results and Analysis

This section provides an in-depth analysis of the results of different macroscopic and microscopic measurements of social networks, together with the observed trends within communities. The subsequent Section 5 further explores and elaborates on the implications of these metrics. For our network analysis, we employed standard Python libraries, including NetworkX, to compute key network metrics such as degree distribution, shortest path distance, network diameter, density, clustering coefficient, CC, and eigenvector centrality. All calculations were performed using default settings. For example, centrality measures were calculated using the built-in functions of NetworkX and igraph, ensuring consistency and accuracy.

4.1. Degree Distribution of InN

The degree distribution of the InN for seven different cuisines adheres closely to a power law, as demonstrated by our analysis. We modeled this behavior using a power law distribution, detailed in equation (1), and visually represented the fit with a red line in Figure 4. In particular, the parameter $γ$ varies between cuisines, ranging from 1.96 in Italian cuisine to 2.38 in Indian cuisine, highlighting the distinct combination of ingredients characteristic of each cuisine. From the linear regression analysis performed on the log-transformed data across 10 global cuisine InNs, we observed a consistent range of values for slope, intercept, $R$ -squared, and $p$ -values. Specifically, the slope ranged from $- 2.45$ to $- 2.68$ , with an intercept between 0.18 and 0.22. The $R$ -squared values remained high, ranging from 0.9965 to 0.9991, indicating an excellent fit between the log-transformed degree sequence and the degree distribution. Furthermore, the $p$ -values were extremely low (ranging from $10^{- 25}$ to $10^{- 30}$ ), providing strong statistical evidence that the slope of the regression line is significantly different from zero. These results consistently support the hypothesis that InNs in diverse cuisines exhibit power-law behavior, reinforcing the robustness of our findings. $p_{k} \sim k^{- γ} .$ (1)

Figure 4.

Degree distribution comparison of seven global cuisines’ ingredient network (InN) along with fitted power law.

4.2. Distance

In graph theory, the distance between two vertices is defined as the number of edges in the shortest path connecting them, often referred to as the hop distance. This concept is mathematically expressed as: $d (V_{i}, V_{j}) = min hopdistance (V_{i}, V_{j}),$ (2)where $d (V_{i}, V_{j})$ represents the minimum number of hops between the vertex $V_{i}$ and the vertex $V_{j}$ . The distance between ingredients in the InN reflects the degree of interconnection and cohesiveness within a cuisine. A shorter maximum distance, such as in Thai cuisine (two steps), suggests a tightly integrated InN, where most ingredients frequently co-occur, forming a highly cohesive flavor structure. In contrast, a larger maximum distance, as observed in Indian cuisine (four steps), indicates a more diverse and modular network, where distinct ingredient groups exist, often used in specialized combinations within different regional or traditional dishes. The fact that most cuisines exhibit a maximum distance of 3 reinforces the small-world nature of culinary networks, where even seemingly unrelated ingredients can be connected through a few intermediate ingredients. This structure enables both traditional recipe formation and culinary innovations, as the short paths between ingredients facilitate cross-category ingredient substitutions and the development of fusion cuisine.

Our analysis of different cuisines revealed varying maximum distances, underscoring the ultra-small world nature of these networks. For example, the maximum distance in the InN of Indian cuisine is 4, indicating a wider variety of ingredient pairings, whereas in Thai cuisine, the maximum distance is only 2, suggesting a tighter integration of ingredients. Other cuisines typically exhibited a maximum distance of 3, further demonstrating the close-knit structure of these culinary networks.

4.3. Diameter

The diameter of a graph is determined by its maximum eccentricity, which is calculated as follows: $\begin{aligned} E (V_{i}) = & max_{V_{j} \in V ∖ V_{i}} (d (V_{i}, V_{j})), \end{aligned}$ (3) $\begin{aligned} D (G) = & max_{V_{i} \in V} (E (V_{i})) . \end{aligned}$ (4)

In these equations, $E (V_{i})$ represents the eccentricity of the vertex $V_{i}$ , defined as the greatest distance from $V_{i}$ to any other vertex $V_{j}$ . The diameter of the graph, $D (G)$ , is then the largest eccentricity among all vertices in the graph. This metric reflects the farthest distance between any two nodes in the network. In particular, in the InN for each InN analyzed cuisine, the diameter is consistently measured 4. The diameter of the InN provides insights into the reachability and connectivity of ingredients within a cuisine. The observation that the diameter remains consistently 4 across all cuisines suggests that InNs are highly interconnected, meaning that even the most distantly related ingredients can be linked through a small number of intermediate connections. This reflects the efficient structure of culinary traditions, where the core ingredients act as bridges connecting diverse groups of ingredients. For example, in Italian cuisine, ingredients such as olive oil and garlic serve as key connectors in various dishes, while in Japanese cuisine, ingredients such as soy sauce and dashi link otherwise distinct flavor profiles. The small diameter highlights the flexibility of culinary systems, allowing ingredients to be combined in innovative ways while maintaining a cohesive flavor network. This structural characteristic underscores the adaptability of cuisines, where a limited number of widely used bridging ingredients facilitate the fusion and evolution of culinary traditions.

4.4. Density

Density in a network measures the proportion of actual edges to the maximum possible edges between nodes, reflecting the network connectivity. The denser a network is, the more interconnected it is, with a large number of edges relative to the number of nodes. The density of the InN varies between different cuisines, with node counts ranging from 784 in British cuisine to 1,926 in Italian cuisine, and edge counts from 14,479 to 62,528, respectively. The density is calculated as follows: $Density = \frac{Actual number of edges}{Maximum possible edges},$ (5)where the maximum possible edges for a graph with $n$ nodes is $(\binom{n}{2})$ . The density of the InN provides important insights into the interconnectivity of ingredients within different cuisines. A higher density suggests that a cuisine has a highly integrated InN, where many ingredients frequently co-occur in diverse recipes, enabling greater flexibility in ingredient combinations. For example, Thai cuisine, with the highest observed density (0.067), features a tightly interconnected InN, reflecting the frequent pairing of core ingredients such as fish sauce, lemongrass, chili, and coconut milk in various dishes. In contrast, Indian cuisine, which has a lower density (0.0296), exhibits a more modular structure, where ingredients, such as spices, form distinct clusters, often used in specialized combinations rather than universally in all dishes. The observed variation in density between cuisines suggests that some culinary traditions rely on highly versatile and broadly connected ingredient sets, while others emphasize distinctive and clustered ingredient groups, reinforcing the unique structural and cultural organization of different cuisines.

4.5. Cluster Coefficient

The clustering coefficient is an essential metric in SNA, as it indicates the degree to which the nodes tend to cluster together, reflecting the formation of communities and shared interests, as highlighted by Katzir and Hardiman (2015). Applied to the InN, this coefficient can identify groups of ingredients that are commonly used together in specific types of recipes. The formula for calculating the clustering coefficient for a node $i$ , which has neighbors $n_{i}$ , is: $C_{i} = \frac{2 e_{i}}{n_{i} (n_{i} - 1)} .$ (6)

Here, $C_{i}$ represents the clustering coefficient for node $i$ , where $e_{i}$ is the number of actual connections between the neighbors of node $n_{i}$ .

The clustering coefficient in the InN provides key insights into the structural organization of the relationships between culinary ingredients. A high clustering coefficient suggests that certain groups of ingredients are frequently used together, forming tightly knit culinary clusters that define the flavor profiles of specific cuisines. For example, the high clustering coefficient observed in Mexican cuisine (0.8399) reflects the strong interconnectedness of ingredients such as chilies, tomatoes, cilantro, and lime, which commonly co-occur in a variety of traditional dishes. Similarly, the French cuisine network (0.7986) indicates well-established ingredient pairings, such as butter, garlic, wine, and thyme, which frequently appear together in classical French cooking. The variation in clustering coefficients across cuisines suggests that some culinary traditions rely on highly modular InNs, where groups of ingredients form cohesive communities that characterize distinct regional flavors. This highlights how different cuisines develop unique ingredient interaction patterns, reinforcing their cultural identity and traditional cooking methods.

4.6. Closeness Centrality

CC) is a critical measure for understanding the proximity of vertices in a graph, which, in the context of the InN, helps identify how readily accessible ingredients are to each other and reveals the flexibility in ingredient usage across recipes. CC is calculated using the formula: $C (V_{i}) = \sum_{V \in G} \frac{1}{d (V_{i}, V_{j})} .$ (7)

A histogram illustrating the CC values for all ingredients is presented in Figure 5, highlighting the range of CC, which typically spans from 0.4 to 0.6 for every cuisine in the InN. This range indicates the degree of closeness and connectivity among ingredients within the various cuisines.

Figure 5.

Closeness centrality distribution for the ingredient network (InN) of Indian cuisine.

The CC of ingredients in the InN provides valuable information on the accessibility and functional versatility of ingredients within a cuisine. Ingredients with higher CC values tend to be widely used and easily combinable, allowing for greater flexibility in recipe formulation. These ingredients often serve as bridge elements, enabling diverse combinations of ingredients and fostering innovation in cooking. For example, ingredients such as onion, salt, butter, and lemon frequently appear in multiple recipes across different cuisines, demonstrating high accessibility and broad culinary adaptability. In contrast, ingredients with lower CC values may be more specialized or region-specific, appearing primarily in select dishes or niche culinary traditions. The observed CC range (0.4 to 0.6) in cuisines suggests that most ingredients maintain a moderate level of accessibility, ensuring a balance between core staple ingredients and specialized components that contribute to the unique identity of a cuisine. This further underscores the structural efficiency of InNs, where highly central ingredients enhance the cohesion and adaptability of a culinary tradition.

4.7. Eigen Centrality (EC)

EC is a key metric for determining the importance of nodes within a network, as it quantifies the influence of a node based on its connections to other highly significant nodes, as outlined by South et al. (2020). In the context of the InN, EC has been applied to assess the significance of each ingredient. The CE values for most ingredients have been found in every cuisine to generally fall between 0.01 and 0.08. However, as shown in Figure 6, there are some ingredients that stand out with notably higher EC values, underscoring their pivotal roles within their respective culinary networks.

Figure 6.

Eigen centrality distribution for the ingredient network.

In the context of culinary patterns, the EC of ingredients offers valuable insights into their functional roles within different cuisines. Ingredients with higher EC values tend to be those that frequently co-occur with other influential ingredients, often serving as essential flavor bases, binding agents, or key enhancers in traditional dishes. For example, in many global cuisines, garlic, onion, and olive oil exhibit high EC values, highlighting their widespread importance across multiple dishes and their strong connections to other significant ingredients. Conversely, ingredients with lower EC values may represent niche or region-specific components that are less interconnected but still contribute to the distinctiveness of a cuisine. The observed EC distribution across cuisines suggests that staple ingredients with high culinary adaptability and cross-linking properties tend to emerge as central nodes in InNs, reinforcing their fundamental role in shaping flavor complexity and regional food identities.

4.8. Community Structure

The InN was analyzed using weighted Leiden (W-Leiden), Louvain (W-Louvain), and WABCD algorithms (Khanna et al., 2023), to identify distinct culinary communities within various cuisines. The results of the W-Leiden algorithm revealed that the number of communities ranged from 3 in Italian cuisine to 8 in Japanese cuisine. In contrast, the W-Louvain algorithm detected between 4 communities in several cuisines such as Chinese, Japanese, Italian, and Thai, and up to nine communities in British cuisine, with Southern US cuisine standing out with 17 communities. The WABCD algorithm showed a variation from six communities in Chinese cuisine to nine communities in Southern US cuisine. These findings underscore the tendency of the InN to form community structures similar to those of other social networks.

Figure 7 presents the community detection results within the InN of Indian cuisine, illustrating the segmentation achieved by the WABCD, W-Leiden, and W-Louvain algorithms, which identified seven, five, and four distinct communities, respectively. The figure clearly shows that the first community identified by each algorithm is characterized by tightly interconnected nodes, indicating strong cohesion. However, the density of nodes and connectivity within communities tend to decrease in subsequent groups identified by each algorithm, reflecting a varying degree of association among different ingredient groups. These patterns suggest underlying structures in the data that mirror complex relationships within the culinary domain.

Figure 7.

Results from different community detection algorithms: (a) WABCD detects seven communities; (b) W-Leiden detects five communities; and (c) W-Louvian detects four communities. Note. WABCD = weighted association-based community detection; W-Leiden = weighted Leiden; W-Louvian = weighted Louvian.

To uncover the inherent characteristics of each partition, we have created multiple subgraphs based on the category of recipes and compared them with the communities obtained from all three algorithms. The results were shown in Tables 2 and 3. One may observe that with both W-Leiden and W-Louvain algorithms, the second community exhibits a connection with recipe category Desert, whereas the rest tend to have more association with Lunch/Dinner category. Conversely, the WABCD identifies four prominent recipe categories: Bread, Lunch/Dinner, Drink, and Desserts.

Table 2.

Cuisine Wise Statistics.

Cuisine	Total number of recipes
Indian	5,187
Chinese	2,673
Japanese	1,423
SouthernUS	4,320
French	2,646
Italian	7,838
Spanish	989
Mexican	6,438
British	4,320
Thai	1,539

Table 3.

Comparison of Community Detection Algorithms.

	Weighted Leiden	Weighted Louvain	WABCD
C1	Lunch/Dinner recipes	Lunch/Dinner recipes	Bread recipes
C2	Dessert recipes	Dessert recipes	Bread recipes
C3	Lunch/Dinner recipes	Lunch/Dinner recipes	Lunch/Dinner recipes
C4	Lunch/Dinner recipes	Lunch/Dinner recipes	Drink recipes
C5	Lunch/Dinner recipes	—	Lunch/Dinner recipes
C6	—	—	Dessert recipes
C7	—	—	Lunch/Dinner recipes

Note. WABCD = weighted association-based community detection.

5. Discussion

Results of the empirical study reveal that InN shows properties that resemble scale-free networks. For example, Figure 4 demonstrates that InN follows a power-law degree distribution with exponents ranging from $γ = 1.96$ to $γ = 2.38$ . These values of $γ$ ensure the ultra-small property (Cohen & Havlin, 2003), that is, the average diameter of the network is minimal, indicating efficient connectivity. In the context of food networks, this range of $γ$ reflects a scale-free structure, where a few ingredients (hubs) are central to many recipes, while most ingredients have limited connections. This heterogeneity is consistent with other real-world networks, such as social networks ( $γ = 2 -- 3$ ), biological networks ( $γ = 2 -- 2.5$ ), and technological networks ( $γ = 2.1 -- 2.4$ ). The slightly lower gamma values in food networks may highlight the unique role of staple ingredients (e.g., salt, onions, or rice) that are ubiquitous across recipes, shaping the culinary structure of cuisines. In contrast, a random network usually has $γ \geq 3$ (Barabási, 2014).

For a reader’s reference, the degree of separation comparison between real networks and InN for Indian cuisine is shown in Table 4. The bold values shows the InN properties of Indian Cusine. The table shows the average and maximum distances of the five real undirected networks and InN. The maximum distance in a network represents the longest shortest path between any two nodes, reflecting the degree of connectivity and integration within the network. In the context of InN, the maximum distance varies across cuisines due to differences in culinary traditions, ingredient usage patterns, and recipe structures. Cuisines with smaller maximum distances (e.g., Italian or French) often rely on a core set of staple ingredients (e.g., olive oil, tomatoes, or butter) that are widely used across recipes, creating a highly interconnected network. This results in shorter paths between ingredients, as most ingredients are linked through these central hubs. Cuisines with larger maximum distances (e.g., Indian) may exhibit greater diversity in ingredient usage, with distinct regional or cultural variations in recipes. This can lead to less overlap between ingredient clusters, resulting in longer paths between certain ingredients. For instance, the use of specialized spices or herbs in specific dishes may create “bridges” that increase the maximum distance. The columns N, L, k, d, and d_max represent the number of nodes, links, average degree, average distance, and maximum distance, respectively. Degrees of separation specify how many hops one must reach from any randomly chosen node to another. For InN, the maximum distance is 4, indicating that every node can be reached from any node through a maximum hop of 4. The average degree of separation is 3.12. This corresponds to the phenomenon of six degrees of separation (de Sola Pool & Kochen, 1978; Milgram, 1967; Travers & Milgram, 1977), which says one node can be reachable through a maximum of six hops (Newman et al., 2006). Although theoretically, this represents the diameter as 6, practically, the average distance stays at 6 or less. The actual diameter sometimes shows higher values because of outliers. For all pairs of Facebook users worldwide and within the United States, the average distance separation is only $3.90$ (Backstrom et al., 2012). However, there are networks such as Power Grid where the average distance between nodes exceeds 10 degrees of separation. Most real-world social networks adhere to the six degrees of separation principle. Such a network does not show a power law distribution.

Table 4.
Comparison of Real World Undirected Networks With Indian Ingredient Network (InN) as to Degree Separation and Fluctuations (Newman et al., 2006).

Network N L d $d_{max}$ $⟨ k ⟩$ $⟨ k ⟩$ $γ$

Internet 192,244 609,066 6.98 26 6.34 240.10 3.42 $^{}$

Power grid 4,941 6,594 18.99 46 2.67 10.30 Exp.

Science collaboration 23,133 93,437 5.35 15 8.08 178.20 3.35 $^{}$

Actor network 702,388 29,397,908 3.91 14 83.71 47,353.70 2.12 $^{}$

Protein interactions 2,018 2,930 5.61 14 2.90 32.30 2.89 $^{ -}$

InN (Indian cuisine) 1433 30,464 3.12 4 1.17 14.69 2.38 $^{*}$

Network	N	L	d	$d_{max}$	$⟨ k ⟩$	$⟨ k ⟩$	$γ$
Internet	192,244	609,066	6.98	26	6.34	240.10	3.42 $^{*}$
Power grid	4,941	6,594	18.99	46	2.67	10.30	Exp.
Science collaboration	23,133	93,437	5.35	15	8.08	178.20	3.35 $^{*}$
Actor network	702,388	29,397,908	3.91	14	83.71	47,353.70	2.12 $^{*}$
Protein interactions	2,018	2,930	5.61	14	2.90	32.30	2.89 $^{* -}$
InN (Indian cuisine)	1433	30,464	3.12	4	1.17	14.69	2.38 $^{*}$

The nodes of InN adhere to an $80 : 20$ ratio, meaning that 80% of the nodes have a low degree and 20% have a higher degree. For InN, most nodes (597) have degrees ranging from 0 to 100, and a smaller number of nodes (156) have degrees ranging from 200 to 500.

Table 4 also shows the degree fluctuations in real networks, indicating the average degree $⟨ k ⟩$ and the second moment $⟨ k 2 ⟩$ for five undirected real networks and InN. Except for the power grid, every network is scale-free as the estimated degree component $γ$ offers a statistically significant fit. The confidence of the fit to the degree distribution is indicated by the star next to the given values.

The InNs for 10 global cuisines shown in Table 5 exhibit distinct structural properties, reflecting the unique culinary traditions, cooking methods, and ingredient co-occurrence patterns in each cuisine. The table provides insights into key network metrics such as network size ( $N$ ), edge count ( $L$ ), power-law exponent ( $γ$ ), maximum degree ( $d_{max}$ ), diameter, density, clustering coefficient, centrality measures, and community structures. Below is a detailed analysis comparing these cuisines based on their network properties and their implications in culinary practices.

Table 5.

Social Metrics Summary of 10 Cuisine’s Ingredient Network.

										Communities
Cuisine wise ingredient network	N	L	$γ$	d_max	Diam.	Dens.	Cluster coefficient	Closeness centrality	Eigen centrality	W-Louvian	W-Leiden	WABCD
Indian	1,433	30,464	2.38	4	4	0.03	0.81	0.40 to 0.60	0.01 to 0.04	4	5	7
Chinese	1,748	62,062	2.12	3	4	0.04	0.83	0.40 to 0.60	0.01 to 0.06	6	4	6
Japanese	959	21,552	2.06	3	4	0.05	0.81	0.40 to 0.60	0.01 to 0.08	8	4	7
Southern US	1,606	42,802	2.04	3	4	0.03	0.81	0.40 to 0.60	0.01 to 0.06	7	17	9
French	1,432	35,028	1.98	3	4	0.03	0.79	0.40 to 0.60	0.01 to 0.06	7	6	7
Italian	1,926	62,528	1.96	3	4	0.03	0.81	0.40 to 0.55	0.01 to 0.04	3	4	8
Spanish	836	16,864	2.10	3	4	0.05	0.81	0.40 to 0.55	0.01 to 0.06	6	5	7
Mexican	1,756	58,972	2.09	3	4	0.04	0.84	0.40 to 0.55	0.01 to 0.04	4	5	8
British	784	14,479	1.99	3	4	0.05	0.80	0.40 to 0.55	0.01 to 0.07	5	9	7
Thai	889	26,531	2.15	2	4	0.07	0.83	0.45 to 0.55	0.01 to 0.06	4	4	7

Note. WABCD = weighted association-based community detection; W-Leiden = weighted Leiden; W-Louvian = weighted Louvian.

First, the network size and connectivity vary significantly across cuisines. Italian cuisine has the largest InN with 1,926 nodes and 62,528 edges, indicating high ingredient diversity and frequent co-occurrence in recipes. Similarly, Indian, Chinese, and Mexican cuisines also have large InNs, reflecting their rich culinary traditions. In contrast, British cuisine has the smallest network (784 nodes, 14,479 edges), suggesting a more limited ingredient repertoire and simpler recipe structures. The power-law exponent ( $γ$ ) ranges from 1.96 to 2.38, with Indian cuisine exhibiting the highest value (2.38), suggesting a strong hierarchical structure dominated by a few essential ingredients.

Second, the small-world properties of these networks demonstrate their connectivity and accessibility of the ingredients. The maximum shortest path length ( $d_{max}$ ) is the lowest for Thai cuisine (2) and the highest for Indian cuisine (4), indicating that Thai cuisine has a tightly connected core set of ingredients, while Indian cuisine exhibits a broader range of pairings of ingredients. Despite variations in $d_{max}$ , the diameter of the network remains consistently 4 in all cuisines, reinforcing the small-world nature of the relationships between ingredients. The density of the networks is highest for Thai (0.07) and Spanish (0.05) cuisines, indicating strong ingredient co-occurrence, while cuisines such as Indian, French, and Southern US (0.03) exhibit more modular structures with diverse ingredient groupings.

Third, the clustering coefficients reveal the tendency of ingredients to form cohesive communities. The highest clustering is observed in the Mexican (0.84) and Chinese (0.83) cuisines, reflecting their strong reliance on foundational ingredient combinations (e.g., chilies, corn, and beans in Mexican; soy sauce, garlic, and ginger in Chinese). In contrast, French cuisine has the lowest clustering coefficient (0.79), suggesting a more diverse range of ingredient combinations without strongly interconnected clusters. The number of communities detected varies widely, with Southern US cuisine showing the highest number of communities (17 in W-Leiden), reflecting the influence of multiple regional culinary traditions.

Fourth, the centrality measures highlight the influence and accessibility of key ingredients in each cuisine. CC (0.40 to 0.60) remains relatively consistent across cuisines, indicating that ingredients are well-connected in most recipes. However, EC varies more significantly; for example, Japanese cuisine has the highest EC (0.01 to 0.08), suggesting a balanced network where multiple ingredients share influence, whereas Indian and Mexican cuisines have lower values (0.01 to 0.04), indicating dominance by a few key ingredients such as spices, chilies, or staple grains.

Finally, these network properties provide valuable insight into cooking and culinary practices in different cultures. Highly clustered cuisines such as Mexican, Thai, and Chinese rely on strong ingredient groups that frequently appear together, reflecting their use of staple sauces, spice bases, and well-defined flavor profiles. In contrast, less clustered cuisines such as French, Indian, and British suggest a more segmented approach, where ingredients are used in distinct recipe contexts. The large and diverse InNs in Italian, Indian, and Chinese cuisine reflect their culinary complexity and regional variations, while tightly connected networks such as Thai and British indicate a more compact set of ingredients frequently used.

Similarly, Figure 8 presents a scatter plot comparing the standard deviation of the degrees of the nodes with the mean degree ( $⟨ k ⟩$ ) for different real-world networks, including the InN, Power Grid, Protein Interactions, Internet, and Science Collaboration networks. Each point represents a specific network and its position on the $x$ -axis corresponds to $⟨ k ⟩$ (the average number of connections per node), while the $y$ -axis represents the standard deviation of the degree distribution.

Figure 8.

An illustration of the comparison of standard deviation in real-world networks with an ingredient network of Indian cuisine.

Low $⟨ k ⟩$ and low standard deviation for InN, power grid, and protein interactions: The InN, power grid, and protein interactions networks have a low mean degree ( $⟨ k ⟩ \approx 3$ ) and a low standard deviation ( $\approx 1.5 -- 1.8$ ). This indicates that most nodes (ingredients, power stations, or proteins) have a similar number of connections, with fewer extreme hubs. In the culinary context of InN, this suggests that most of the ingredients in the network have a balanced number of connections with other ingredients, reflecting the structured nature of the recipes.

Higher $⟨ k ⟩$ and standard deviation for internet and science collaboration networks: the internet and science collaboration networks have much higher $⟨ k ⟩$ (between 6 and 8) and a greater standard deviation (above 2.5). This means that these networks have many highly connected hub nodes, which are significantly more connected than the average node. The internet follows a scale-free structure in which a few nodes (high-traffic websites) dominate connectivity. The science collaboration network shows that some researchers have an exceptionally high number of collaborations, creating a hub-like structure. The low standard deviation in InN suggests that most ingredients are relatively similar in terms of their frequency of usage in recipes. Unlike the internet, where some nodes (websites) dominate, there are no extreme hubs in InN, which means that recipes are not overly dependent on a few highly connected ingredients. The low $⟨ k ⟩$ value ( $\sim 3$ ) means each ingredient connects to a small number of other ingredients on average, highlighting the different groups of ingredients common in culinary traditions. This aligns with the idea that cuisines have sets of staple ingredients that frequently appear together rather than an arbitrary combination of all available ingredients. Since InN does not have an extremely high variance in connectivity, it suggests that most ingredients can be substituted with similar alternatives, maintaining the structure of the network. This is particularly important in culinary practices, where ingredients can be swapped based on regional availability, dietary restrictions, or personal preferences.

The structural differences observed in these global InNs align with their respective culinary traditions, historical influences, and regional variations. Cuisines with high clustering and density reflect cohesive ingredient usage, while those with diverse community structures exhibit more varied and modular culinary patterns. These insights not only help to understand the fundamental structure of cuisines, but also offer potential applications in food science, recipe recommendation systems, and gastronomy research.

6. Applications

Understanding whether InNs exhibit social properties is significant for both food computing and network science. By analyzing their structure and comparing them with real-world social systems, we gain valuable insight into how ingredients interact and cluster based on culinary traditions, regional preferences, and flavor compatibility. The small-world and scale-free properties of InNs further reinforce their social nature, with profound implications beyond theoretical significance.

6.1. Based on Small-World Property

However, the small-world property, characterized by high clustering and short average path lengths, indicates that InNs are highly interconnected and efficiently organized. This can be used in recipe recommendation systems, where understanding the proximity and relationships of ingredients enhances the accuracy and diversity of recommendations. Ingredients within the same community or cluster are likely to share similar flavor profiles, facilitating better substitutions of ingredients and suggestions of complementary pairings.

6.2. Based on Scale-Free Network

Similarly, the scale-free nature of InNs, with some highly connected hub ingredients, highlights the critical role of these hubs in culinary systems. These staple ingredients serve as foundational elements in many recipes, making them essential to predict culinary trends. Furthermore, the robustness of scale-free networks to random failures suggests that culinary systems can adapt to the removal of less central ingredients, informing strategies for ingredient substitution and dietary adaptation. Using these network properties, our work advances the field of food computing, offering a foundation for developing innovative applications such as flavor pairing prediction, trend analysis, personalized recipe generation, and new recipe generation. Furthermore, demonstrating social behavior in InNs extends SNA techniques to food systems, opening new avenues for interdisciplinary research in computational gastronomy.

7. Conclusion and Future Work

In this paper, we conducted an empirical study to demonstrate that a food InN exhibits structural properties similar to social networks, such as scale-free behavior, community structure, and centrality hierarchies. Through a thorough examination of 10 diverse cuisines, we showed that ingredient networks (InNs) consistently follow a power-law distribution and display social network-like patterns, as evidenced by the computation of relevant metrics. These findings suggest that ingredient interactions in culinary systems are governed by principles similar to those observed in social networks, such as preferential attachment and modular organization.

While our study primarily focuses on the structural analysis of InNs, the observed properties provide a strong foundation for future research into culinary trends and user interactions. For example, the scale-free nature of InNs implies that certain ingredients act as hubs, playing a critical role in the composition of the recipe, which could inform the prediction of the popularity of the ingredients or the evolution of culinary practices. Similarly, the community structure of InNs highlights ingredient groupings that reflect cultural or flavor-based affinities, offering insight into how users might interact with or perceive different cuisines. However, these would require additional empirical studies, such as analyzing temporal data on recipe creation or user preferences, to validate their feasibility.

Our work establishes a robust framework for understanding the organizational dynamics of InNs and opens new avenues to explore their applications in culinary science and user behavior analysis. Future research could build on these findings to develop predictive models or investigate the cultural and psychological factors driving ingredient co-occurrence in recipes.

Footnotes

Funding

The authors disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: Suman Kundu received funding through grant no. 4(2)/2024-ITEA of MeitY,Government of India;and from Srijan,Center for Generative AI (grant no. ET/23/2024-ET) of MeitY under the IndiaAI mission with the support of Meta for partial support.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Data Availability Statement

The INDoRI dataset used in this study is available online at

ORCID iDs

Sandeep Khanna

Suman Kundu

Chiranjoy Chattopadhyay

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