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e5277d2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | "measure_id","dimension","indicator","legacy_alias","symbol","formula_latex","text_definition","direction","unit","evidence_status","data_available","data_path","visualization_available","visualization_path","reference_keys"
"active_participants","Participation","Active observed participants","num_nodes (when the node set is a period's active addresses)","N_t","N_t=|V_t|","Number of distinct observed position-holder addresses active in period t; not a count of people or independent economic actors.","Higher means broader observed participation, conditional on the address caveat.","position-holder address","observed_descriptive","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv","yes","paper/figures/fig05_real_v5_cross_chain.tex","ao2023aave"
"entrant_count","Participation","Entrants","new nodes","E_t","E_t=|V_t\setminus V_{t-1}|","Addresses active in period t but not in t-1.","Higher means more observed address entry; it need not mean new economic actors.","position-holder address","observed_descriptive","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv","no","","holme2012temporal"
"exit_count","Participation","Exits","departing nodes","X_t^{-}","X_t^{-}=|V_{t-1}\setminus V_t|","Addresses active in t-1 but absent in t.","Higher means more observed address exit.","position-holder address","observed_descriptive","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv","no","","holme2012temporal"
"retention","Participation","Retained participants and retention rate","retained nodes","R_t","R_t=|V_t\cap V_{t-1}|;\quad r_t=R_t/|V_{t-1}|","Addresses appearing in two adjacent periods and their share of the prior-period set.","Higher r_t means greater address persistence, not necessarily actor persistence.","position-holder address / share","observed_descriptive","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv","no","","holme2012temporal"
"effective_participants","Participation","Effective number of participants","inverse HHI","N^{\ast}","N^{\ast}=H^{-1};\quad H=\sum_i s_i^2","Number of equal-share participants that would generate the observed concentration.","Higher means a more even activity distribution.","effective address count","observed_derived","yes_derived","outputs/real_v5/descriptive/weekly_address_metrics.csv;outputs/real_v4/ethereum/period_beneficiary_bounds.csv","no","","hirschman1964paternity"
"cross_chain_jaccard","Participation","Cross-chain address-set overlap","Jaccard overlap","J_t","J_t=\frac{|V_{E,t}\cap V_{A,t}|}{|V_{E,t}\cup V_{A,t}|}","Overlap of Ethereum and Arbitrum position-holder-address sets at the same chain-relative event week.","Higher means more repeated addresses across panels; because calendar dates differ it is not contemporaneous migration.","share","observed_descriptive","yes","outputs/real_v5/descriptive/cross_chain_overlap.csv","no","","jaccard1901distribution"
"multilayer_participation","Participation","Multilayer participation coefficient","cross-layer participation","P_i","P_i=1-\sum_{\ell}(k_{i\ell}/k_i)^2","Dispersion of node i's connections across network layers.","Higher means connections are spread across more layers.","node score in [0,1]","planned_blocked","no_verified_multilayer_edges","","no","","kivela2014multilayer"
"total_activity","Transaction","Total observed activity","total_value / event_count","X_t","X_t=\sum_i x_{it}","Total action count in the empirical panels or equilibrium activity in the simulation; heterogeneous token amounts are not summed across reserves.","Scale measure, not a decentralization score by itself.","events or model activity","observed_and_synthetic","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv;outputs/simulation/scenario_results.csv","yes","paper/figures/fig03b_simulation_outcomes.tex","ao2023aave"
"activity_hhi","Transaction","Pool-event-frequency Herfindahl--Hirschman index","activity_hhi","H_t","H_t=\sum_i s_{it}^2;\quad s_{it}=x_{it}/\sum_jx_{jt}","Squared-share concentration of equally weighted Pool events assigned to the stated observed unit.","Higher means greater event-frequency concentration; address HHI is not actor HHI or value concentration.","index in [1/n,1]","observed_visualized","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv;outputs/real_v4/ethereum/period_beneficiary_bounds.csv","yes","paper/figures/fig05_real_v5_cross_chain.tex;paper/figures/fig04_real_v4_partial_identification.tex","hirschman1964paternity"
"actor_hhi_identified_set","Transaction","Economic-actor HHI period bounds and change envelope","entity-adjusted HHI","H_t^a,H_t^g,\mathcal B_\Delta","H_t^a\leq H_t^g\leq1;\quad \mathcal B_\Delta\subseteq[H_1^a-1,1-H_0^a]","Period-specific actor-HHI bounds are sharp; subscripts 0 and 1 denote pre and post periods. Subtracting them gives a conservative outer envelope unless one common controller partition is optimized jointly.","An outer envelope crossing zero does not prove that both signs are jointly feasible.","period bounds / outer envelope","observed_partially_identified","yes","outputs/real_v4/ethereum/period_change_bounds.csv","yes","paper/figures/fig04_real_v4_partial_identification.tex","molinari2020partial;binette2022entity"
"normalized_entropy","Transaction","Normalized Shannon entropy","entropy","S_t","S_t=-\frac{\sum_i s_{it}\log s_{it}}{\log n_t}","Evenness of the activity-share distribution.","Higher means more even activity.","index in [0,1]","planned_not_reported","derivable_not_governed","outputs/real_v5/descriptive/weekly_address_metrics.csv","no","","shannon1948theory"
"gini","Transaction","Gini coefficient","activity_gini","G_t","D_t=\sum_{ij}|x_{it}-x_{jt}|;\quad G_t=D_t/(2n_tX_t)","Mean pairwise inequality in observed activity; the numerator aggregates pairwise absolute differences and the activity total provides the normalization.","Higher means greater inequality.","index in [0,1]","observed_descriptive","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv","no","","gini1912variabilita"
"top_share","Transaction","Top-p activity share","top1pct_share / top10pct_share","T_{p,t}","T_{p,t}=\sum_{j=1}^{\lceil pn_t\rceil}s_{(j)t}","Share held by the largest p fraction of observed units; p is 1% or 10% in the locked panel.","Higher means greater top-tail concentration.","share","observed_descriptive","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv","no","","ao2023aave"
"nakamoto_51","Transaction","Minimum 51% coalition","nakamoto_51","K_{.51,t}","S_{k,t}=\sum_{j=1}^{k}s_{(j)t};\quad \mathcal K_t=\{k:S_{k,t}\geq.51\};\quad K_{.51,t}=\min\mathcal K_t","Smallest number of observed units whose cumulative ordered share reaches 51%; the calligraphic K indexed by t is the admissible threshold set.","Higher means control of the measured activity is spread across more units.","address count","observed_descriptive","yes","outputs/real_v5/descriptive/weekly_address_metrics.csv","no","","zhang2022sokdecentralization"
"chain_hhi","Transaction","Chain concentration","chain_hhi","H_{\ell}","H_{\ell}=\sum_{\ell}s_{\ell}^{2}","Concentration of activity across chain layers.","Higher means activity is concentrated on fewer chains.","index","synthetic_only","synthetic","outputs/simulation/scenario_results.csv","yes","paper/figures/fig03b_simulation_outcomes.tex","kivela2014multilayer"
"graph_size","Structural","Graph nodes and edges","num_nodes / num_edges","n,m","n=|V|;\quad m=|E|","Size of the address-role graph and number of unique observed ties.","Scale measures; neither alone ranks decentralization.","nodes / edges","observed_descriptive","yes","outputs/real_v5/topology/address_role_topology_metrics.csv","yes","paper/figures/fig06_real_v5_core_periphery.tex","ao2023aave"
"density","Structural","Graph density","Relative degree","\delta_G","\delta_G=\chi m/[n(n-1)];\quad \chi\in\{1,2\}","Fraction of possible ties that are observed; chi equals 1 for a directed graph and 2 for an undirected graph.","Direction is context-dependent; density normally falls mechanically as sparse graphs grow.","share","legacy_benchmark","derivable_not_governed","locked row-level Pool events","no","","newman2018networks"
"components","Structural","Weak components and giant-component share","Components_cnt / giant_com_ratio","C,g","C=|\mathcal C(G)|;\quad g=|V^{\star}|/n","Number of disconnected weak components and the fraction of nodes in the largest one; the component family is C(G) and the largest node set is V-star.","Larger g means greater reachability; it is not necessarily greater decentralization.","count / share","partly_observed","yes_for_giant_share","outputs/real_v5/topology/address_role_topology_metrics.csv","no","","newman2018networks;ao2023aave"
"reciprocity","Structural","Directed reciprocity","directed_reciprocity","r","r=m_{\leftrightarrow}/m","Fraction of directed edges that participate in a reciprocated dyad.","Higher means more bidirectional address-role interaction; economic meaning depends on edge semantics.","share","observed_descriptive","yes","outputs/real_v5/topology/address_role_topology_metrics.csv","no","","garlaschelli2004reciprocity"
"degree_moments","Structural","Degree moments and top-hub ratio","Degree mean / Degree std / Top10Degree mean / Top10Degree std / Top10 Degree mean ratio","\bar d,\sigma_d,Q_{10}","\bar d=n^{-1}\sum_i d_i;\quad \sigma_d^2=n^{-1}\sum_i(d_i-\bar d)^2;\quad Q_{10}=\bar d_{10}/\bar d","Average and dispersion of observed neighbors and the top-ten hub multiple; d-bar-10 is the mean degree of the ten highest-degree nodes.","Higher dispersion or Q10 indicates a more unequal degree distribution.","degree / ratio","legacy_benchmark","derivable_not_governed","locked row-level Pool events","no","","freeman1979centrality;ao2023aave"
"degree_centrality_moments","Structural","Degree-centrality mean and dispersion","DCmean / DCstd","\bar c,\sigma_c","c_i^d=d_i/(n-1);\quad \sigma_c^2=n^{-1}\sum_i(c_i^d-\bar c)^2","Normalized degree and cross-node dispersion used in the original benchmark.","Higher dispersion indicates greater hub inequality.","centrality / dispersion","legacy_benchmark","derivable_not_governed","locked row-level Pool events","no","","freeman1979centrality;ao2023aave"
"strength_hhi","Structural","Weighted out- and in-strength HHI","weighted_out_degree_hhi / weighted_in_degree_hhi","H^{\rightarrow},H^{\leftarrow}","p_i^{\rightarrow}=w_i^{\rightarrow}/\sum_jw_j^{\rightarrow};\quad H^{\rightarrow}=\sum_i(p_i^{\rightarrow})^2;\quad p_i^{\leftarrow}=w_i^{\leftarrow}/\sum_jw_j^{\leftarrow};\quad H^{\leftarrow}=\sum_i(p_i^{\leftarrow})^2","Concentration of action-side and position-holder-side Pool-event weights; the arrowed quantities are the corresponding normalized strength shares.","Higher means more event weight is concentrated on fewer address roles.","index","observed_descriptive","yes","outputs/real_v5/topology/address_role_topology_metrics.csv","no","","hirschman1964paternity"
"clustering_transitivity","Structural","Local clustering moments and transitivity","Cluster_mean / Cluster_std / Transitivity","\bar C,\sigma_C,T","C_i=2e_i/[d_i(d_i-1)];\quad T=3\Delta_G/\Lambda_G","Local and global closure of the undirected projection; Delta-G counts triangles and Lambda-G counts connected triples.","Higher means greater triadic closure; it does not have a universal decentralization sign.","coefficient","legacy_benchmark","derivable_not_governed","locked row-level Pool events","no","","watts1998collective;ao2023aave"
"spectral_distance_centrality","Structural","Eigenvector and closeness moments","eig_mean / eig_std / closeness_mean / closeness_std","\mathbf c,c_i^{\ell}","A\mathbf c=\lambda_1\mathbf c;\quad c_i^{\ell}=(n-1)/\sum_{j\neq i}d_{ij}","Recursive-neighbor importance and inverse geodesic distance, summarized by their mean and dispersion.","Higher dispersion indicates concentration of the stated centrality; disconnected-graph conventions must be reported.","centrality","legacy_benchmark","derivable_not_governed","locked row-level Pool events","no","","freeman1979centrality;newman2018networks"
"pagerank","Structural","Weighted PageRank and concentration","maximum_pagerank_share / pagerank_hhi","\boldsymbol\pi,H_{\pi}","\mathbf q=(1-\eta)\mathbf 1/n;\quad \boldsymbol\pi=\mathbf q+\eta P^\top\boldsymbol\pi;\quad H_{\pi}=\sum_i\pi_i^2","Stationary importance under weighted directed transitions and its concentration; eta is the damping parameter and q is the teleport vector.","Higher PageRank HHI means PageRank mass is concentrated on fewer nodes.","node score / index","observed_visualized","yes","outputs/real_v5/topology/address_role_topology_metrics.csv;outputs/real_v5/core_periphery/node_coreness.csv.gz","yes","paper/figures/fig06_real_v5_core_periphery.tex","brin1998anatomy"
"modularity","Structural","Community modularity","Modularity","Q","B_{ij}=A_{ij}-k_ik_j/(2m);\quad \delta_{ij}=\mathbf 1\{g_i=g_j\};\quad Q=(2m)^{-1}\sum_{ij}B_{ij}\delta_{ij}","Excess within-community connectivity relative to the configuration benchmark; the residual edge term is weighted by a binary same-community indicator.","Higher means a stronger partition, not automatically more decentralization.","score","planned_not_reported","no_governed_output","","no","","newman2006modularity;traag2019leiden"
"k_core","Structural","Core number, maximum k-core, and maximum-core share","maximum_k_core / maximum_core_node_share","c_i,k_{\star},S_{\star}","c_i=\max\{k:i\in G_k\};\quad I_i^{\star}=\mathbf 1\{c_i=k_{\star}\};\quad S_{\star}=n^{-1}\sum_i I_i^{\star}","Nested degree-based cohesion in the unweighted undirected projection; the maximum nonempty shell is paired with a binary membership indicator.","Higher coreness means deeper cohesion; the maximum shell and its share must be read together.","node shell / count / share","observed_visualized","yes","outputs/real_v5/core_periphery/node_coreness.csv.gz;outputs/real_v5/core_periphery/summary.json","yes","paper/figures/fig06_real_v5_core_periphery.tex","batagelj2003cores"
"borgatti_everett","Structural","Borgatti--Everett binary core and fit","cp_test_pvalue / cp_significance / core_cnt / avg_core_neighbor (original); current release reports fit and core membership","b_i,\rho_B","B_{ij}=\mathbf 1\{b_i+b_j\geq1\};\quad \rho_B=\rho(A,B)","Fits a dense core with core--periphery ties and sparse periphery--periphery ties; the fit is the correlation between observed and ideal adjacency, and the current audit requires at least two core nodes.","Larger fit means closer agreement with the binary ideal; core count is definition-sensitive.","binary node class / fit","observed_visualized","yes","outputs/real_v5/core_periphery/node_coreness.csv.gz;outputs/real_v5/core_periphery/summary.json","yes","paper/figures/fig06_real_v5_core_periphery.tex","borgatti2000models;kojaku2018core"
"rombach_coreness","Structural","Rombach continuous coreness","continuous coreness robustness","\chi_i,R_{\alpha\beta}","R_{\alpha\beta}=\sum_{ij}A_{ij}C_iC_j;\quad \chi_i\propto\sum_{\alpha,\beta}C_i(\alpha,\beta)R_{\alpha\beta}","Quality-weighted continuous coreness aggregated over the locked alpha-beta grid.","Higher means a stronger continuous core position under this definition.","node score in [0,1]","observed_visualized","yes","outputs/real_v5/core_periphery/node_coreness.csv.gz;outputs/real_v5/core_periphery/summary.json","yes","paper/figures/fig06_real_v5_core_periphery.tex","rombach2014core"
"core_persistence","Structural","Temporal maximum-core persistence","persistent coreness","P_i^{k}","P_i^{k}=\frac{\sum_t\mathbf 1\{c_{it}=k_{\star t}\}}{\sum_t\mathbf 1\{i\in V_t\}}","Fraction of a node's active topology weeks spent in that week's maximum shell.","Higher means a more persistent address-role core position.","share","observed_data_ready","yes","outputs/real_v5/core_periphery/node_coreness.csv.gz","no","","holme2012temporal"
"betweenness_brokerage","Structural","Betweenness brokerage","brokerage","b_i","b_i=\sum_{s\neq i\neq t}\sigma_{st}(i)/\sigma_{st}","Share of shortest paths between other nodes that pass through i.","Higher means greater geodesic brokerage under the chosen path weights.","node score","planned_not_reported","derivable_but_not_governed","locked row-level Pool events","no","","freeman1979centrality"
"community_persistence","Structural","Temporal community persistence","community persistence","P_{g,t}^{\mathcal C}","J_{gh,t}=\frac{|C_{g,t}\cap C_{h,t+1}|}{|C_{g,t}\cup C_{h,t+1}|};\quad P_{g,t}^{\mathcal C}=\max_h J_{gh,t}","Best adjacent-period Jaccard match for a detected community.","Higher means more stable community membership, conditional on the algorithm.","share","planned_not_reported","no_governed_community_output","","no","","holme2012temporal;traag2019leiden"
"multiplier_centrality","Structural","Network multiplier centrality and structural HHI","structural_hhi","z,H_z","z=(I-\alpha G)^{-1}\mathbf 1;\quad H_z=\sum_i(z_i/\sum_jz_j)^2","Model-implied amplification of activity through network complementarities and its concentration.","Higher Hz means multiplier influence is more concentrated.","model score / index","synthetic_only","synthetic","outputs/simulation/scenario_results.csv","yes","paper/figures/fig03b_simulation_outcomes.tex","ballester2006keyplayer"
"route_hhi","Infrastructure","Route HHI and effective routes","route_hhi","H_R,N_R^{\ast}","H_R=\sum_k q_k^2;\quad N_R^{\ast}=H_R^{-1}","Concentration of verified interchain activity across independent routes.","Higher route HHI means greater route concentration; a higher effective route count means more dispersion.","index / effective route count","synthetic_only_empirical_blocked","synthetic_only","outputs/simulation/scenario_results.csv","yes_synthetic_only","paper/figures/fig03b_simulation_outcomes.tex","belchior2021survey"
"facilitator_share","Infrastructure","Bridge or facilitator share","bridge/facilitator shares","q_k","q_k=x_k/\sum_jx_j","Share of verified interchain activity assigned to infrastructure k.","Higher maximum share means greater single-infrastructure dependence.","share","empirical_blocked","no_verified_route_event_table","","no","","belchior2021survey"
"removal_loss","Infrastructure","Targeted route-removal loss","max_route_removal_loss","L_k","L_k=(X-X^{(-k)})/X","Fraction of feasible activity lost when infrastructure k and its dependent links are removed.","Higher means greater functional dependence on k.","share","synthetic_only_empirical_blocked","synthetic_only","outputs/simulation/scenario_results.csv","yes_synthetic_only","paper/figures/fig03b_simulation_outcomes.tex","albert2000error"
"edge_disjoint_paths","Infrastructure","Edge-disjoint route redundancy","disjoint paths","\lambda_{uv}","\lambda_{uv}=|\mathcal P_{uv}^{\star}|","Cardinality of a maximum family of pairwise edge-disjoint routes between the relevant endpoints.","Higher means more topological redundancy, conditional on verified route edges.","path count","empirical_blocked","no_verified_route_event_table","","no","","ford1956maximal"
"minimum_cut","Infrastructure","Minimum cut","minimum cut","\kappa_{uv}","\mathcal S_{uv}=\{S:u\in S,v\notin S\};\quad \kappa_{uv}=\min_{S\in\mathcal S_{uv}}\sum_{e\in\delta(S)}w_e","Smallest verified route capacity whose removal separates the two endpoints; the calligraphic set enumerates all separating node partitions.","Higher means greater cut-based redundancy.","capacity or edge count","empirical_blocked","no_verified_route_event_table","","no","","ford1956maximal"
"expected_failure_loss","Infrastructure","Expected infrastructure-failure loss","expected route loss","\mathcal L","\mathcal L=\sum_k\pi_kL_k","Failure-probability-weighted activity loss across infrastructure components.","Higher means greater expected infrastructure exposure.","expected share","model_only_not_estimated","no_empirical_failure_probabilities","","no","","acemoglu2015systemic"
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