Foundational Principles of Social Network Analysis

Building on the formal graph-theoretic definition of social networks as ordered pairs $G = (V, E)$ where $V$ denotes the set of nodes (actors, organizations, or entities) and $E$ denotes the set of edges (social ties, interactions, or relational flows), this section elaborates on the typological classification of ties and their operationalization, as the nature of ties fundamentally determines the structural properties of the network and the validity of subsequent analytical results. Tie typologies are not merely descriptive categorizations: they map directly to the mathematical properties of the graph, the selection of appropriate centrality metrics, and the interpretability of diffusion and influence outcomes. We first formalize tie classification along four orthogonal dimensions: symmetry, weight, valence, and multiplexity, before deriving core network metrics and reviewing data collection and ethical frameworks for expert research.

1.1 Tie Typologies and Operational Measurement

The first dimension of tie classification is symmetry, which distinguishes undirected ties from directed ties. An undirected tie represents a symmetric relationship where the existence of a tie from node $v_i$ to $v_j$ implies a reciprocal tie from $v_j$ to $v_i$; classic examples include mutual friendship, co-membership in a voluntary association, or shared organizational affiliation. Formally, an undirected edge is an unordered pair $\{v_i, v_j\} \in E$, and the adjacency matrix $A$ of an undirected graph is symmetric ($A_{ij} = A_{ji}$ for all $i,j$). A directed tie, by contrast, represents an asymmetric relationship where the flow of resources, information, or affect is unidirectional: examples include following on social media, citing a scholarly work, transferring funds, or reporting to a supervisor in a hierarchical organization. Directed edges are ordered pairs $(v_i, v_j) \in E$, and the adjacency matrix is not necessarily symmetric. The choice between directed and undirected modeling is not trivial: modeling a directed citation network as undirected, for example, would erase the directional flow of scholarly influence, leading to biased estimates of structural prestige.

The second dimension is weight, which distinguishes weighted ties from unweighted ties. An unweighted tie is a binary indicator of tie existence ($A_{ij} = 1$ if a tie exists, 0 otherwise), while a weighted tie assigns a scalar value $w_{ij} \geq 0$ to each edge representing the strength, frequency, or intensity of the relationship. Tie weight is most commonly operationalized via three measurable dimensions: (1) interaction frequency (e.g., number of in-person meetings per month, number of digital messages exchanged per week); (2) affective intensity (e.g., self-reported closeness on a Likert scale from 1 to 10); and (3) instrumental exchange (e.g., number of times the two nodes have provided resources or favors to each other). For example, a study of interorganizational collaboration networks in the U.S. biotech industry weighted ties by the number of joint research projects between firms, rather than treating all collaborative ties as identical, as project volume is strongly correlated with knowledge transfer volume[^2].

The third dimension is valence, which distinguishes positive, negative, and neutral ties. Positive ties represent affiliative, cooperative, or supportive relationships (e.g., friendship, trust, collaboration), while negative ties represent antagonistic, competitive, or hostile relationships (e.g., enmity, distrust, rivalry). Signed social networks, which incorporate both positive and negative ties, have been shown to explain a range of outcomes that unsigned networks cannot, including group conflict, political polarization, and organizational sabotage. For example, a study of U.S. Senate cosponsorship networks found that incorporating negative ties (i.e., pairs of senators who never cosponsor bills together and belong to opposing parties) improved the accuracy of predicting legislative gridlock by 34% relative to unsigned network models[^3]. Neutral ties represent the absence of any relationship, and are typically encoded as 0 in adjacency matrices.

The fourth dimension is multiplexity, which distinguishes single-mode ties from multiplex ties. A single-mode tie represents a single type of relationship between two nodes (e.g., only a work relationship, only a friendship), while a multiplex tie represents multiple overlapping relationship types between the same pair of nodes (e.g., a pair of nodes who are both coworkers and friends, or who are both family members and business partners). Multiplex ties are common in real-world social systems: a study of small business owners in rural Kenya found that 62% of collaborative business ties overlapped with kinship or friendship ties, and that multiplex ties were 2.7 times more likely to survive a financial shock than single-mode ties[^4]. Operationalizing multiplexity requires either (1) modeling multiple edge types as separate layers in a multilayer network, or (2) aggregating tie strength across relationship types into a single weighted edge, with the choice dependent on the research question.

> 【电子版侧边栏】强连接与弱连接的经典测量框架 > Granovetter’s (1973) seminal work on the strength of weak ties defined tie strength along four orthogonal, measurable dimensions: (1) interaction frequency (number of contacts per unit time); (2) emotional intensity (self-reported closeness, typically on a 1–10 scale); (3) relationship duration (years of acquaintance); and (4) reciprocal exchange (frequency of mutual resource provision). Empirical studies typically operationalize tie strength as a weighted sum of these four dimensions, with scores above 6 classified as strong ties and below 3 as weak ties; scores between 3 and 6 are classified as ambiguous or moderate ties. This framework has been validated across more than 200 cross-cultural studies, with consistent predictive validity for outcomes including job search success, information diffusion, and social support provision.

1.2 Formal Definitions and Mathematical Properties of Core Network Metrics

Network metrics can be divided into two categories: node-level metrics, which quantify the structural position of individual actors, and network-level metrics, which quantify the global structural properties of the entire network. We derive formal definitions for the four most widely used node-level metrics, followed by two core network-level metrics, with a worked example using a 5-person startup founding team to illustrate calculation.

1.2.1 Node-Level Centrality Metrics

Degree centrality is the simplest and most intuitive node-level metric, quantifying the number of direct ties a node has. For an undirected, unweighted graph, the raw degree centrality of node $v_i$ is $deg(v_i) = \sum_{j=1}^n A_{ij}$, where $n$ is the total number of nodes in the network. To enable cross-network comparison, degree centrality is typically standardized by dividing by the maximum possible degree ($n-1$ for simple graphs without self-loops): $$C_D(v_i) = \frac{deg(v_i)}{n-1}$$ For directed graphs, degree centrality is split into in-degree (number of incoming ties, e.g., followers, citations received) and out-degree (number of outgoing ties, e.g., followings, citations sent): $$C_D^{in}(v_i) = \frac{deg^{in}(v_i)}{n-1}, \quad C_D^{out}(v_i) = \frac{deg^{out}(v_i)}{n-1}$$ For weighted graphs, degree centrality is extended to strength centrality, which sums the weights of all edges incident to $v_i$: $$C_S(v_i) = \frac{\sum_{j=1}^n w_{ij}}{n-1}$$ Degree centrality is most useful for identifying highly active or popular nodes in a network, but it does not account for the structural position of the node’s ties: a node with high degree centrality may only connect to other nodes within a single isolated community, with no access to information or resources outside that community.

Betweenness centrality quantifies the ex