# Self-Complexity Measurement Specification v2.1 ## Everythingist Research Dashboard — Revised April 2026 --- ## Overview This document specifies all self-complexity metrics implemented in the Research Dashboard, including: - Legacy metrics (Scott, 1969; applied by Linville, 1985, 1987) - Component metrics (Rafaeli-Mor et al., 1999) - Composite metric (Sakaki, 2004) - Spatial/AST metrics (reconstructed from Schleicher & McConnell, 2005) - Connectivity and network metrics (novel) - Novel SSM extensions that formalize identity as a geometric-network system Self-complexity is theoretically defined as a function of: 1. Number of self-aspects 2. Degree of independence vs. overlap among those aspects (Linville, 1985, 1987) --- ### Changes from v1.0 | Change | Detail | |--------|--------| | **Scott H** (renamed) | Formerly "Linville H." The H statistic was created by Scott (1969); Linville adopted it. Display key: `scottH`. | | **OL citation** | Now cites Rafaeli-Mor et al. (1999, Appendix B) with Tversky (1977) asymmetry rationale. | | **Sakaki SC** | Attribution corrected to Sakaki (2004), who introduced the SC = K/OL composite. Components from Rafaeli-Mor et al. (1999). | | **Sakaki SC null** | Returns null when OL = 0 (mathematically undefined). No epsilon-stabilized variant. Standard practice: exclude from analysis. | | **Singleton handling** | W_k = null (not 0) when |A_k| < 2. Derived metrics propagate null. New tracking fields: `singletonAspectCount`, `validAspectCount`. | | **AST label** | Explicitly labeled as "reconstructed operationalization" — not a canonical published formula. | | **Volume → Distributedness** | Renamed. Not a geometric volume; scales with K. Dashboard key: `distributedness`. | | **Balance** | Normalized version is the only version. Raw removed. Dashboard key: `balanceIndex`. | | **Dashboard keys** | Every metric now has an explicit dashboard key mapping and tooltip documentation. | | **References** | Added Scott (1969), Sakaki (2004), Tversky (1977), Jaccard (1912), Newman (2006), Shannon (1948). | --- ## Data Structure and Conventions Each participant provides: - $K$ self-aspects - A set of attributes/traits - Each aspect $A_k$ contains a subset of attributes - Attributes may appear in multiple aspects Optional (for spatial metrics): - Attribute coordinates in $\mathbb{R}^d$ ### Global Notation - $K$ = number of self-aspects - $K^*$ = number of valid aspects (≥2 attributes) — used for W-dependent metrics - $A_k$ = set of attributes in aspect $k$ - $|A_k|$ = number of attributes in aspect $k$ - $X$ = set of all unique attributes used across aspects - $|X|$ = number of unique attributes - $x_{ka}$ = coordinate vector for attribute $a$ in aspect $k$ - $\mu_k$ = centroid of aspect $k$ - $d(\cdot,\cdot)$ = distance metric (default: Euclidean) - $\epsilon = 10^{-6}$ (recommended stabilizer for denominators) ### Singleton Aspect Handling (Critical — New in v2) When $|A_k| < 2$: - $W_k$ = **null** (not 0) - Aspect is excluded from W-dependent metric computation - Track: `singletonAspectCount`, `validAspectCount` - If ALL aspects are singletons ($K^* = 0$) → W-dependent metrics return null **Rationale:** Setting $W_k = 0$ for singletons artificially inflates Crystallization, IRI, Isolation Ratio, and AST ratios. Null propagation is the mathematically honest approach. **Affected metrics:** Within Spread (W), AST SC, Crystallization, Isolation Ratio, IRI. --- # A. LEGACY METRICS ## 1. Number of Self-Aspects $$ NSA = K $$ Where: - $K$ = total number of self-aspects > **In plain language:** How many distinct "hats" you wear — e.g., professor, parent, athlete, musician. Each one is a self-aspect. **Dashboard key:** `nsa` --- ## 2. Scott H $$ H = \log_2(n) - \frac{\sum_i n_i \log_2(n_i)}{n} $$ Where: - $n$ = total number of attributes available for sorting - $n_i$ = number of attributes in unique group combination $i$ ### Attribution The H statistic was developed by Scott (1969) as a measure of the number of independent dimensions underlying a cognitive sorting. Linville (1985, 1987) adopted it as her primary operationalization of self-complexity. The field conventionally called it "Linville's H," but the mathematics are Scott's. ### Notes - A **group combination** is the pattern of aspect membership for a trait - Combinations with $n_i = 0$ are excluded ### Interpretation - Higher $H$ → more differentiation in sorting patterns - BUT: strongly influenced by number of selected traits - Does **not cleanly measure overlap or independence** - Poor internal consistency when valence is considered (Rafaeli-Mor et al., 1999) - Confounds number of aspects with overlap — this is the central critique that motivated Rafaeli-Mor et al.'s component decomposition > **In plain language:** How differently you sort your traits across roles. If every role uses the same traits, H is low. If each role draws from its own unique mix, H is high. Think of it as measuring how varied your trait-sorting patterns are. **Dashboard key:** `scottH` **Citation:** Scott, 1969; applied to self-complexity by Linville, 1985, 1987 --- # B. COMPONENT METRICS ## 3. Overlap (OL) — Directed Formulation $$ OL = \frac{\sum_i \left(\sum_j \frac{C_{ij}}{T_j}\right)}{K(K-1)} $$ Where: - $C_{ij}$ = number of shared attributes between aspects $i$ and $j$ - $T_j$ = total attributes in the referent aspect $j$ - $K$ = number of aspects - $i$ and $j$ vary from 1 to $K$ ($i \neq j$) - Division by zero: skip pair if $T_j = 0$ ### Attribution Formalized by Rafaeli-Mor et al. (1999, Appendix B) as a standalone component metric. The directed (asymmetric) formulation follows Tversky (1977, p. 328): "similarity should not be treated as a symmetric relation." All ordered pairs $(i, j)$ where $i \neq j$ are evaluated, producing $K(K-1)$ comparisons (not $\binom{K}{2}$). Rafaeli-Mor et al. demonstrated that OL has robust split-half reliability even under valence-based splitting (β = .55–.56), unlike Scott H which drops to near-zero. > **In plain language:** How much your roles recycle the same traits. If "disciplined" shows up in every role you define, overlap is high. Low overlap means each role has its own distinctive character. **Dashboard key:** `overlapOL` **Citation:** Rafaeli-Mor et al., 1999; directed formulation per Tversky, 1977 --- ## 4. Pairwise Overlap (Jaccard — Auxiliary) $$ OL_{ij} = \frac{|A_i \cap A_j|}{|A_i \cup A_j|} $$ $$ \bar{J} = \frac{2}{K(K-1)} \sum_{i **In plain language:** Of all the traits two roles use *combined*, what fraction do they share? If your professor-self and athlete-self together use 20 traits but only 2 appear in both, that's low overlap (0.10). This is measured for every pair of roles, then averaged. **Dashboard key:** `meanJaccard` **Citation:** Jaccard, 1912; applied per Rafaeli-Mor et al., 1999 --- # C. COMPOSITE METRIC ## 5. Sakaki Self-Complexity $$ SC = \frac{NSA}{OL} $$ ### Edge Case If $OL = 0$: $$ SC = \text{null (undefined)} $$ Division by zero is mathematically undefined. The standard practice in the literature is to exclude participants with zero overlap from SC analysis. **No epsilon-stabilized variant is provided.** The v1 stabilized version ($SC_\epsilon = K / (OL + \epsilon)$) has been removed — it was an engineering convenience, not a methodological choice. ### Attribution The composite SC = NASPECTS/OL was introduced by Sakaki (2004, p. 129), who cited Rafaeli-Mor et al. (1999) for the component measures. Importantly, Rafaeli-Mor et al. proposed studying NSA and OL as *separate* components and explicitly argued against collapsing them into a single index. Sakaki combined them into a ratio for use as a predictor of mood-incongruent recall. > **In plain language:** A ratio of how many roles you have to how much they overlap. Many roles with little recycling = high complexity. Few roles that all look the same = low complexity. Undefined when there is zero overlap (every role uses entirely unique traits). **Dashboard key:** `sakakiSC` **Citation:** Sakaki, 2004; components from Rafaeli-Mor et al., 1999 --- # D. SPATIAL / AST METRICS **Important:** These metrics are a **reconstructed operationalization** based on Schleicher & McConnell (2005), reflecting distance between self-aspect centroids relative to within-aspect spread. They are NOT an exact reproduction of a published formula. ## 6. Aspect Centroid $$ \mu_k = \frac{1}{m_k} \sum_{a=1}^{m_k} x_{ka} $$ Where: - $m_k$ = number of attributes in aspect $k$ - $x_{ka}$ = coordinate of attribute $a$ > **In plain language:** The "center of gravity" of a role — where it lives in trait-space. If your athlete-self is defined by "competitive," "driven," and "energetic," the centroid is the average position of those three traits. --- ## 7. Between-Aspect Separation $$ B = \sum_{i **In plain language:** How far apart your roles are from each other overall. A professor-self and an athlete-self that use very different traits will be far apart; two roles built from similar traits will be close together. This sums all those distances. **Dashboard key:** `betweenSeparation` --- ## 8. Mean Separation $$ \bar{B} = \frac{2}{K(K-1)} \sum_{i **In plain language:** The average distance between any two roles. Same idea as total separation, but adjusted so it doesn't grow just because you have more roles. **Dashboard key:** `meanSeparation` --- ## 9. Within-Aspect Spread $$ W_k = \frac{1}{\binom{m_k}{2}} \sum_{a **In plain language:** How diverse the traits *within* a single role are. A role defined by just "competitive" and "driven" is tight and focused. One spanning "creative," "anxious," "nurturing," and "bold" is spread out. $W$ is the average spread across all your roles. Singleton aspects (only 1 attribute) are excluded — they don't have enough data to compute internal spread. **Dashboard key:** `withinSpread` **Additional keys:** `singletonAspectCount`, `validAspectCount` **Citation:** Reconstructed from Schleicher & McConnell, 2005 --- ## 10. AST Self-Complexity $$ SC_{AST} = \frac{B}{W + \epsilon} $$ Returns null when $W$ = null (all aspects are singletons). > **In plain language:** Are your roles far apart relative to how spread out each one is internally? Think of it as signal-to-noise for identity separation. High values mean your roles are clearly distinct from one another; low values mean the differences between roles are small compared to the messiness within them. **Dashboard key:** `astSC` --- ## 11. Normalized AST Complexity $$ SC_{AST}^{norm} = \frac{\bar{B}}{W + \epsilon} $$ Returns null when $W$ = null. > **In plain language:** Same idea as AST complexity, but using average (rather than total) separation. This makes the metric comparable across people who have different numbers of roles. **Dashboard key:** `astSCnorm` **Citation:** Reconstructed operationalization based on Schleicher & McConnell, 2005 --- # E. DERIVED STRUCTURAL METRICS These metrics are derived from the spatial quantities above to capture coherence, scale, and independence of the identity system. **Novel metrics developed by Sean P. Mullen, PhD, University of Illinois Urbana-Champaign.** ## 12. Crystallization $$ Crystallization = \frac{1}{W + \epsilon} $$ Returns null when $W$ = null (all aspects are singletons). ### Interpretation - High → internally coherent, tightly defined aspects - Low → diffuse, loosely organized aspects > **In plain language:** How tightly defined your roles are. A person with crisp, clear-cut identities (narrow trait clusters) scores high. Someone whose roles feel fuzzy and amorphous — traits scattered everywhere — scores low. **Dashboard key:** `crystallization` --- ## 13. Distributedness $$ D = K \cdot \bar{B} $$ ### Interpretation - Higher → expansive identity space (many, well-separated aspects) - Lower → constrained identity space ### Limitations - Not a true geometric "volume" — scales linearly with K - Should not be interpreted as physical space - Captures breadth of identity distribution that complexity indices (H, SC) do not isolate > **In plain language:** How much total "identity real estate" you occupy. Many well-separated roles = high distributedness, like rooms spread across a big house. Few overlapping roles = low distributedness, like everything crammed into one studio apartment. **Dashboard key:** `distributedness` --- ## 14. Isolation Ratio $$ Isolation = \frac{\bar{B}}{W + \epsilon} $$ Returns null when $W$ = null. ### Interpretation - High → aspects are far apart relative to their internal spread - Low → aspects bleed into one another > **In plain language:** How independent your roles feel from one another. High means your work-self and your parent-self barely touch — what happens in one stays in one. Low means they blur together. **Dashboard key:** `isolationRatio` --- ## 15. Identity Rigidity Index (IRI) $$ IRI = \frac{1}{K^*} \sum_{k : |A_k| \geq 2} \frac{1}{W_k + \epsilon} $$ Returns null when $K^* = 0$ (all aspects are singletons). Singleton aspects excluded from summation. ### Interpretation - High $IRI$ → tight, well-defined (rigid) aspects - Low $IRI$ → diffuse, loosely defined aspects ### Limitations - Inflated by small within-aspect variance - Sensitive to number of attributes per aspect > **In plain language:** How locked-in each role is. Rigid roles have narrow, fixed trait profiles — you always describe your athlete-self with the exact same words. Flexible roles are loosely defined and might shift depending on context. **Dashboard key:** `rigidityIndex` --- # F. CONNECTIVITY METRICS These metrics capture how attributes bridge across aspects, driving cross-identity spillover. **Novel metrics developed by Sean P. Mullen, PhD, University of Illinois Urbana-Champaign.** ## 16. Bridge Load For each attribute $x$: $$ BridgeLoad(x) = \sum_{k=1}^{K} \mathbf{1}(x \in A_k) $$ > **In plain language:** For a given trait, how many roles does it connect? "Empathy" appearing in your parent-self, therapist-self, and friend-self has a bridge load of 3. It's a bridge connecting three parts of your identity. --- ## 17. Mean Bridge Load (Bridge Centrality) $$ BC_{mean} = \frac{1}{|X|} \sum_{x \in X} BridgeLoad(x) $$ ### Interpretation - High values → many cross-aspect connectors (greater spillover pathways) - Low values → more isolated aspects ### Limitations - Depends on attribute assignment process - Sensitive to trait redundancy — identical words across aspects drive this up > **In plain language:** On average, how many roles does each trait touch? If most of your traits only appear in one role, this is low. If your traits tend to show up across many roles, this is high — meaning there are lots of pathways for experiences in one role to ripple into others. **Dashboard key:** `meanBridgeLoad` --- ## 18. Proportion Shared Attributes $$ PropShared = \frac{|\{x \in X : BridgeLoad(x) > 1\}|}{|X|} $$ > **In plain language:** What fraction of your traits show up in more than one role? If 15 out of 30 traits appear in multiple roles, PropShared = 0.50. High values mean your identity is highly interconnected; low values mean your roles are built from mostly unique traits. **Dashboard key:** `propShared` --- # G. NETWORK METRICS These metrics treat the identity system as a graph, with aspects as nodes and attribute overlap as edges. ## 19. Spillover Risk Index (SRI) $$ SRI = \frac{2}{K(K-1)} \sum_{i **In plain language:** If something bad happens in one role, how likely is it to bleed into others? This is driven by roles that share many traits *and* sit close together in trait-space. A bad day at work is more likely to ruin your evening at home if your work-self and parent-self share a lot of the same traits and feel closely connected. **Dashboard key:** `spilloverRisk` **Novel metric:** Developed by Sean P. Mullen, PhD, University of Illinois Urbana-Champaign. Captures a risk dimension that pure overlap or pure distance metrics miss. --- ## 20. Identity Modularity (Q) ### Graph construction - Nodes = aspects - Edge weight = $w_{ij}$ (e.g., Jaccard overlap) Let: - $A_{ij}$ = weighted adjacency matrix - $k_i = \sum_j A_{ij}$ (node strength) - $m = \frac{1}{2}\sum_{ij} A_{ij}$ (total edge weight) - $c_i$ = community assignment for node $i$ (via community detection) - $\delta(c_i, c_j)$ = 1 if same community, else 0 ### Newman Modularity $$ Q = \frac{1}{2m} \sum_{ij} \left( A_{ij} - \frac{k_i k_j}{2m} \right)\delta(c_i, c_j) $$ ### Interpretation - High $Q$ → well-separated identity modules - Low $Q$ → entangled / blended identity structure > **In plain language:** Do your roles form natural clusters — like a "work cluster" (professor, researcher, mentor) and a "home cluster" (parent, partner, friend) — or is everything tangled together? High modularity means clean clusters with clear boundaries between them. ### Implementation notes - Use a weighted community detection algorithm (e.g., Louvain) to obtain $c_i$ - If $m = 0$ (no edges), set $Q = 0$ (no modular structure) - Resolution limit for very small networks ($K < 5$) **Dashboard key:** `modularity` **Citation:** Newman, 2006 (modularity); Louvain method for community detection --- # H. DISTRIBUTIONAL METRICS These metrics characterize how evenly the identity system allocates resources across aspects. ## 21. Identity Entropy Let: $$ p_k = \frac{|A_k|}{\sum_{j=1}^{K} |A_j|} $$ Then: $$ H_{identity} = -\sum_{k=1}^{K} p_k \log(p_k) $$ ### Interpretation - High entropy → evenly distributed identity composition - Low entropy → dominance of one/few aspects > **In plain language:** Are your roles evenly sized, or does one dominate? A person who's 90% "work-self" and 10% everything else has low entropy — most of their identity eggs are in one basket. Someone who invests equally across roles has high entropy, meaning a more balanced portfolio. ### Normalized version $$ H_{norm} = \frac{H_{identity}}{\log(K)} $$ Range: 0–1. Use the normalized version for cross-participant comparison (raw entropy scales with K). **Dashboard keys:** `identityEntropy`, `identityEntropyNorm` **Citation:** Shannon, 1948 (information entropy) --- ## 22. Balance Index (Normalized) Using $p_k$ from the entropy section: $$ Balance = 1 - \frac{\mathrm{std}(p_k)}{\mathrm{std}_{max}} $$ Where $\mathrm{std}_{max}$ is the maximum possible standard deviation for a $K$-category distribution. ### Interpretation - High balance → evenly weighted identities (approaching 1.0 = perfect equality) - Low balance → dominance imbalance across aspects ### Notes This IS the normalized version. Not comparable across different K without normalization — and this metric already includes it. The raw variant ($1 - \sigma$) has been removed. > **In plain language:** Same intuition as entropy but measured differently: how equal is the "weight" you give to each role? If one role has 15 traits and the rest have 2 each, balance is low. If all roles are roughly the same size, balance is high. **Dashboard key:** `balanceIndex` **Novel metric:** Developed by Sean P. Mullen, PhD, University of Illinois Urbana-Champaign. --- # Interpretation Summary | Metric | Key | Section | Attribution | Plain Language | |--------|-----|---------|-------------|----------------| | NSA | `nsa` | A | Linville, 1985 | How many hats you wear | | Scott H | `scottH` | A | Scott, 1969; Linville, 1985 | How differently you sort traits across roles | | OL | `overlapOL` | B | Rafaeli-Mor et al., 1999 | How much roles recycle the same traits | | Mean Jaccard | `meanJaccard` | B | Jaccard, 1912 | Symmetric pairwise overlap | | Sakaki SC | `sakakiSC` | C | Sakaki, 2004 | Many unique roles vs. few similar ones | | B | `betweenSeparation` | D | Schleicher & McConnell, 2005* | Total distance between all role pairs | | B̄ | `meanSeparation` | D | Schleicher & McConnell, 2005* | Average distance between role pairs | | W | `withinSpread` | D | Schleicher & McConnell, 2005* | How diverse traits are within a role | | SC_AST | `astSC` | D | Schleicher & McConnell, 2005* | Signal-to-noise for identity separation | | SC_AST_norm | `astSCnorm` | D | Schleicher & McConnell, 2005* | Normalized AST (cross-participant) | | Crystallization | `crystallization` | E | Mullen (novel) | How tightly defined each role is | | Distributedness | `distributedness` | E | Mullen (novel) | Total identity real estate | | Isolation | `isolationRatio` | E | Mullen (novel) | Do roles stay in their own lanes? | | IRI | `rigidityIndex` | E | Mullen (novel) | How locked-in each role's trait profile is | | Bridge Load | `meanBridgeLoad` | F | Mullen (novel) | How many roles a trait connects | | Prop Shared | `propShared` | F | Mullen (novel) | Fraction of traits in multiple roles | | SRI | `spilloverRisk` | G | Mullen (novel) | Likelihood a bad day bleeds across roles | | Modularity | `modularity` | G | Newman, 2006 | Do roles form clean clusters? | | Entropy | `identityEntropy` | H | Shannon, 1948 | Are roles evenly sized? | | Entropy (norm) | `identityEntropyNorm` | H | Shannon, 1948 | Normalized evenness (0–1) | | Balance | `balanceIndex` | H | Mullen (novel) | Equity of identity weighting | \* Reconstructed operationalization, not canonical formula. --- # Methodological Notes - Self-complexity includes BOTH number and independence of aspects - The Scott $H$ statistic is widely used but **psychometrically limited** — confounds NSA with OL - Overlap and number should be analyzed separately (Rafaeli-Mor et al., 1999) - Spatial (AST) metrics better capture theoretical structure than $H$ alone - Attribute sharing drives spillover across identities - Default distance: Euclidean; alternatives (city-block) should be logged - Guard denominators with $\epsilon$ only where explicitly noted — do NOT silently stabilize primary values (e.g., Sakaki SC returns null, not $K/\epsilon$) - For singleton aspects: $|A_k| < 2$ → set $W_k$ = null, exclude from W-dependent metrics, track singleton count - Log-transform or z-score metrics for visualization if needed - Report normalized versions for cross-participant comparison (Scott H, Entropy, Balance, AST_norm) --- # Reporting Guidelines When reporting results: - Do NOT collapse these into a single scalar index - Present metrics as a **profile** organized by dimension: - Quantity (NSA) - Overlap (OL / Jaccard) - Geometry (B, W, $SC_{AST}$) - Structure (Crystallization, Distributedness, Isolation, IRI) - Connectivity (Bridge metrics, SRI) - Network (Q) - Distribution (Entropy, Balance) - Emphasize that outcomes likely depend on **interactions among dimensions** - Report singleton aspect counts and any null-returning metrics with sample sizes - Note the Sakaki SC exclusion count when OL = 0 --- # Citation Note These metrics extend prior frameworks: - Scott (1969) — H statistic (cognitive dimensionality) - Linville (1985, 1987) — foundational self-complexity theory and buffering hypothesis - Rafaeli-Mor et al. (1999) — component decomposition (NSA + OL), psychometric critique of H - Sakaki (2004) — composite SC = K/OL formulation - Schleicher & McConnell (2005) — distance-based associated systems approach - McConnell (2011) — multiple self-aspects framework (networked self-concept) The present metrics (Sections E–H) formalize identity as a **geometric-network system**, extending the spatial intuition of Schleicher & McConnell into structural, connectivity, and distributional dimensions. **Novel metrics (Sections E–H) developed by Sean P. Mullen, PhD, University of Illinois Urbana-Champaign.** --- # References Jaccard, P. (1912). The distribution of the flora in the alpine zone. *New Phytologist, 11*(2), 37–50. Linville, P. W. (1985). Self-complexity and affective extremity: Don't put all of your eggs in one cognitive basket. *Social Cognition, 3*(1), 94–120. Linville, P. W. (1987). 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