GLOSSARY

Adjacency is the graph theoretic expression of the fact that two entities, represented by nodes, are directly related, tied, or connected with one another. Formally, given entities ni and nj in a set of agents N, and the A ={aij} arcs denoting the existence of relations from i to j; i and j are adjacent if there exist either of the two arcs, aij or aji. Given the digraph D = (N,A), its adjacency matrix A(D) is defined by A(D) = {aij} where aij=1 if either aij or aji exists, and 0 otherwise.

a density Criterion: a procedure for tie assignment in the construction of a blockmodel. According to this criterion, an arc (bAB) between two blocks (A,B) for a given relation is 1 if the observed density of arcs between the two blocks (dAB) is at least as large as a, and zero otherwise. Often, a is chosen to equal (D), the density of the original sociomatrix. Formally, for a = D; bAB = 1 if dAB ³ D else bAB = 0.

arc: in graph theory, an arc is a directional relation that has an explicit origin and destination, represented by a line between entities with an arrowhead at the destination.

backward linkage = transaction link(s) to the buying sector of interest from industries that supply intermediate goods or services. The supplying industries are also known as 'upstream' sectors in the flow of goods from extraction through raw, processed, and final retailed form to consumers. The magnitude of a sector's backward linkages are given by its column coefficients in the multiplier matrix.

betweenness, a measure of the global centrality of the actor i, is the probability that the shortest path from actor j to actor k takes a route through agent i. Formally, B(ni) = å jN gjk(ni)/gjk where gjk(ni) is the total number of geodesics through i, and 1/gjk is the probability that the particular geodesic is chosen.

blockmodel is an aggregated sociomatrix representation of a network. A blockmodel is created from an elemental sociomatrix data base by (i) partitioning the entities into discrete subsets, called "blocking," and (ii) assigning 0/1 ties between each pair of blocked subsets (White, Boorman and Breiger, 1976; Holland and Leinhardt, 1979).

closeness is a second measure of the global centrality of the actor i. C(ni) = [å jN d(ni,nj)]-1 where d(ni,nj) is the geodesic (shortest path) distance between i and j entities in the network, and N is the network size. To normalize, divide the sum of geodesics by N-1 before inverting.

complement: generally, an input or consumer good whose use decreases as the cost of another rises is considered a complement in demand to that other good. An output or product whose supply decreases as the price of another falls is considered a complement in supply to the other product. Complements generally occur in fixed proportions in an economy. For Keystone Sector Analysis: entities I and J are perfect complements when K's interaction with I is always accompanied by K's interaction with J (Kilkenny and Nalbarte-this paper).

A complete graph is one in which all the actors have two-way ties to all other actors.

A component is the largest subset of actors in a network that all relate to each other, also known as group or a sub-graph. A strong component is one in which the arcs that make up the paths are aligned in continuous chain without a change of direction. A weak component is made of actors that are linked by non-directional edges (Scott, 1991).

cut-point is the node whose removal from the system would increase the number of components by dividing the graph into 2 or more separate components, between which there are no ties (Scott, 1991).

density is the measure of how many entities are related to others in a set. Density (D) is measured by the ratio of the actual number of non-reflexive arcs in proportion to the maximum possible number of non-reflexive arcs: D = å iå j aij / N(N-1) .

dichotomous relation is a tie between two agents (dyad) that either does or does not exist. It is recorded as a binary variable with (1) indicating the presence and (0) indicating the absence of the relation between the two set entities in the dyad.

directed graphs or digraphs are the graphic representations of directional relational data among entities in a set. Entities are illustrated as nodes, and the directional relations, if they exist, are illustrated as arcs, with the arrowhead pointing from the source or sending node to the destination or receiving node. Formally, a digraph is a finite, non-empty set N, whose elements ni = {n1,n2,...ng} are called nodes, together with a set A = {a12, a13,...a1g,...ag-1,g} of ordered pairs aij, called arcs, where ni and nj are distinct members of N (Robinson and Foulds, 1980).

directional relation is an interaction between entities that is specific about which entity is the source versus the sink, which is the origin or destination. It is represented graphically by an arc, a line between entities with an arrowhead at the destination. The sociomatrix for a directional relation will not generally be symmetric, unless all ties are reciprocated.

dominant industry: the sector(s) with relatively high location quotients, that also have a number of input-output linkages with other local industries with high location quotients Cella 1984

A dyad consists of a pair of actors and the possible ties between them (Wasserman and Faust, 1994).

edge: in graph theory, this represents a non-directional relation or tie which is non-specific about the origin or destination of the flow on the link. It is illustrated by a line between the interacting agents that has no arrowhead.

efficient path test is applied to a blockmodel of a community to determine if a sector is a keystone. If the excision of the sector from the image matrix (or reduced graph) reduces the closeness measures for the remaining sectors, that sector is a keystone. (Kilkenny and Nalbarte, this paper)

forward linkage the transaction link(s) from the supplying sector of interest to industries that demand intermediate goods or services. The using industries are also known as 'downstream' sectors in the flow of goods from extraction through raw, processed, and final retailed form to consumers. The magnitude of a sector's forward linkages are given by its row coefficients in the multiplier matrix.

fracture test is a cut-point test based on ties applied to a blockmodel of a community to determine if a sector is a keystone. If the excision of the sector from the image matrix (or reduced graph) destroys the connectivity of the network by increasing the number of components, that sector is a keystone. (Kilkenny and Nalbarte, this paper)

geodesic d(ni,nj) is the shortest path between two nodes i and j. It is measured as the number of arcs required to get from i to j, which is the first power p for which the ijth element of Ap is non-zero: d(ni,nj) = min{p}| AijP >0 (Wasserman and Faust, 1994). Equivalently, for any digraph with adjacency matrix A, each cell aijp of Ap equals the number of paths of length p from node i to j in A, for any positive integer p (Robinson and Foulds, 1980).

image matrix is the aggregated form of an elemental sociomatrix, representing the blockmodel in a matrix comprised of [0,1] cell entries.

immiserizing growth is expansion in a region's real economic output that earns less rather than more real income. This outcome can arise only under two conditions (1) the region must be large in the market for the product, where 'large' means that changes in the region's supply of the product will affect market prices, and (2) demand for the product must be highly inelastic, such that revenues fall even as more is sold.

indegree measures the strength of a node as a sink in a system. It is the number of arcs ending at the node, measured by the column sum for the node in a dichotomous sociomatrix: formally the indegree of actor j = åi aij .

keystone sector is the type of entity (business, institution, organization, etc) in a community that plays a unique role and without which the community is fundamentally and detrimentally altered (Kilkenny, 1997). Based on the terms Keystone Species and turnkey or key sector.

keystone species is the species responsible for the structure and integrity of an ecosystem. The term was first coined by ecologists in the late 1960s (Paine, 1969).

local centrality is a measure of prominence which reflects the number of direct transmissions from the entity, measured by the outdegree (or row sum) for the entity. Also known as degree centrality.

local prestige is a measure of prominence which reflects the number of the entity's direct receipts, measured by the indegree (or column sum) of the entity. Also known as degree prestige. (Wasserman and Faust, 1994).

location quotient: a measure of relative concentration in an area. Given a local proportion, such as the percentage of total region r employment in industry i (eir), and a reference proportion, such as the share of nationwide employment in industry i (eir), the location quotient (LQir) is eir/ei., more formally: LQir = (eir/å i eir)/( åreir/åiåreir) .

multiplier matrix is an array of numbers which show the amount change in the row sector due to a unit change in the column sector. It is calculated from an industry-by-industry transactions matrix in three steps as follows. First, divide the transaction cells by their column totals. This gives a matrix of sectoral expenditure shares [A]. Second, subtract this matrix from the Identity matrix [I-A]. Third, invert this subtrahend. The result is the multiplier matrix "m". Formally, m = [I-A]-1 .

nodes represent the individual entities or actors in networks.

non-directional relation is an interaction between entities that is non-specific about which entity is the source versus the sink, or which is the origin or destination. It is represented graphically by an edge or tie, a simple line between the entities, and by a symmetric sociomatrix.

Oneblock Criterion: a procedure for tie assignment in the construction of a blockmodel. According to this criterion, an arc (bAB) between two blocks (A,B) for a given relation is 1 only if all possible arcs (tij) from all actors in the row block to actors in the column block exist, otherwise the block arc is 0. Formally, bAB = 1 if tij =1 for all i Î A and all j Î B ; else bAB = 0.

Outdegree measures the strength of a node as a source in a system. It is the number of arcs beginning at a node, measured in dichotomous sociomatrix data as the row sum for the node: outdegree of actor i = åj aij

reduced digraph the illustration of arcs between the blocks of nodes for a blockmodel. See also image matrix.

reflexive tie is the relation that a particular entity has with itself. In a sociomatrix, a reflexive tie by the ith entity is recorded by aii = 1. In a digraph, reflexive ties are drawn as arrows that originate and end on the same node, that are curved back on themselves.

A relation is the collection of ties of a specific kind among a set of entities. Alternatively, consider the mathematical definition of binary relation (Robinson and Foulds,1980):

Given two sets S and T, each member of set S may be related to a number (perhaps zero) of members of set T. The mathematical description of this situation is called a binary relation. If s S and tT, then (s,t) is a member of this set when s is related to t.

Sinks Substitute Criterion: a procedure for tie assignment in the construction of a blockmodel. According to this criterion, an arc (bAB) between two blocks (A,B) for a given relation is 1 if there is an arc (tij) from every actor in the row block to at least one actor in the column block, otherwise the block tie is 0. Formally, (Kilkenny and Nalbarte, this paper)

social embeddedness in traditional societies, economic life is submerged ('embedded') in social relations. For example, businesses employ locals and buy locally, no matter what, because it is expected of them by society. In contrast in modern life, social relations are often an epiphenomenon of the market (Granovetter, 1985). For example, when businesses employ locals or buy locally simply because it is profitable, this is the reverse of social embeddedness.

A social network consists of a finite set of actors and the relation or relations defined on them. Actors are social entities, discrete individuals, corporate or collective social units. (Wasserman and Faust, 1994).

sociomatrix is an entity-by-entity array of data on the relational ties between them. Rows of the sociomatrix represent the sending actors while the columns represent the receiving actors. See also digraph.

Sources Substitute Criterion: a procedure for tie assignment in the construction of a blockmodel. According to this criterion, an arc (bAB) between two blocks (A,B) for a given relation is 1 if there is an arc (tij) to every actor in the column block from at least one actor in the column block, otherwise the tie is 0. Formally: .(Kilkenny and Nalbarte, this paper)

structurally equivalent entities have exactly the same directional ties (arcs) to and from all other entities with whom either has ties. Formally, entities i and j are structurally equivalent if, for all other actors k = 1,2,...N; k¹i,j, and all the relations r = 1,2,..., R, arik iff arjk and arki iff arkj .

substitute: generally, an input or consumer good whose use increases as the cost of an alternative rises is considered a substitute in demand for that alternative. An output or product whose supply increases as the price of an alternative falls is considered a substitute in supply to the alternative product. Substitutes generally satisfy similar demands (play similar roles) in an economy. For Keystone Sector Analysis: entities I and J are perfect substitutes if a slight decrease in the desirability of K interacting with I leads to K interacting with J instead (Kilkenny and Nalbarte, this paper).

tie is a relation between two entities in a network. If it has direction it is an arc, if non-directional it is an edge.

transitive a relation R is transitive over the set {X} if for all xi,j or k in {X}, if xi R xj AND xj R xk, then xi R xk. For example, let R be "like" and {X} be people. Like is a transitive relation if every time a person A likes B, and person B likes C, then person A likes C. It is easy to find a counterexample to prove that in fact, "like" is not a transitive relation for people.

turnkey or key sector sectors whose structure of backward and forward linkages create above-average impacts on the rest of the economy. Activities having the highest linkages are considered key sectors because it is thought that concentrating resources in them will stimulate more production, income and employment than alternative allocations of resources. (Cella, 1984). See also dominant industry.

valued relation is a tie that has a magnitude that reflects the level, intensity, or frequency of a relation.

Zeroblock Criterion: a procedure for tie assignment in the construction of a blockmodel. According to this criterion, a tie (bAB) is assigned between two blocks (A,B) for a given relation is 0 only if there are no arcs (tij = 0) from any actor (i) in the row block to any actor (j) in the column block; otherwise the block arc is 1. Formally, bAB = 0 if tij =0 for all i Î A and all j Î B else bAB = 1.